diff --git a/AGENTS.md b/AGENTS.md index f0020b5d..f32b0f0b 100644 --- a/AGENTS.md +++ b/AGENTS.md @@ -769,6 +769,7 @@ fn main() -> genoxide::Result<()> { - Test problems: `multi::problems::{Zdt1, Zdt2, Zdt3, Zdt4, Zdt6}::new(n)`, `Zdt5` (a `Binary` genome of 80 bits, not in `all()`), `{Dtlz1, …, Dtlz7}::::new(n)` (the technical report's numbering), `{Wfg1, …, Wfg9}::::new(k, l)` (k position parameters, a multiple of M − 1; l distance parameters, even for WFG2 and WFG3; `default()`: k = 4 for 2 objectives, 2(M − 1) for more, l = 20), `{FonsecaFleming, Kursawe}::new(n)`, `Schaffer1`, `Schaffer2`, `Poloni`, `Viennet1`/`2`/`3` (3 objectives), and the constrained `Bnh`, `Srn`, `Tnk`, `Osy`, `Constr` (fitness `([f64; 2], violation)`), all minimized, for `MultiEngine::new(algorithm, problem)`. The `multi::problems::MultiProblem` trait gives `representation()`, `optimal_front(points)` (`Option`: `None` for KUR, POL, VNT2, VNT3, DTLZ5, DTLZ6 with 4 or more objectives, and WFG3 with 3 or more), `ideal_point()`, `nadir_point()`, `constraints(&x)` (`g <= 0`), `reference()`; `multi::problems::all::()` lists them as `Box>`. - Constrained test problems of tunable difficulty: `multi::problems::{Ctp1, …, Ctp8}` (two objectives, two variables, disconnected fronts, fronts of separate points, infeasible bands and tunnels) and `{C1Dtlz1, C1Dtlz3, C2Dtlz2, ConvexC2Dtlz2, C3Dtlz1, C3Dtlz4}::::new(n)` (Jain and Deb's constrained DTLZ; `C1Dtlz3` and `ConvexC2Dtlz2` take the paper's radius for 3, 5, 8, 10 and 15 objectives, else `with_radius(n, r)`), all with `optimal_front`, and in `all()`. - More test problems, with their fronts: Deb and Jain's `{ConvexDtlz2, ScaledDtlz1, ScaledDtlz2, InvertedDtlz1}::::new(n)` (scaled: the paper's factor for M, or `.with_factor(s)`), and Ma and Wang's constrained `Mw1`, `Mw2`, `Mw3`, `Mw5`, `Mw6`, `Mw7`, `Mw9`, …, `Mw13` (2 objectives, `new(n)`, n = 15 by default) and `{Mw4, Mw8, Mw14}::::new(n)` (M + 12 by default), all in `all::()`. +- Advanced constrained suites, with their fronts, all in `all::()`: the DTLZ report's `{Dtlz8, Dtlz9}::::new(n)` (10M by default; DTLZ8 from 3 objectives), Li et al.'s DC-DTLZ `{Dc1Dtlz1, Dc1Dtlz3, Dc2Dtlz1, Dc2Dtlz3, Dc3Dtlz1, Dc3Dtlz3}::::new(n)` (`with_parameters(n, a, b)`), and Fan et al.'s `DasCmop1` to `DasCmop9` (2 objectives to `DasCmop6`, 3 from `DasCmop7`; `new(n, difficulty)`, `with_difficulty(d)`, 30 variables and the paper's figure 6 triplet by default), whose `Difficulty::new(η, ζ, γ)` or `Difficulty::standard(1..=16)` (the paper's table 3) sets the constraints' difficulty; their fronts are sampled per triplet. - Engineering designs with several objectives, `multi::problems::engineering::{TwoBarTruss, WeldedBeam, DiscBrake, SpeedReducer, FourBarTruss}` (2 objectives), `{CarSideImpact, RocketInjector, VehicleCrashworthiness, MarineDesign}` (3) and `WaterResourcePlanning` (5), in `all::()`: the trusses' and `WaterResourcePlanning`'s fronts are derived; the others' `optimal_front` is `None`, with an `ideal_point()` (and `nadir_point()` for 2 objectives). `DiscBrake` and `SpeedReducer` round their integer gene; `design(&x)` gives the rounded design. Python: `gx.problems.multi_engineering`. - `multi::indicator`: `hypervolume(&front, &reference_point, &objectives)`, `hypervolume_contributions`; `igd_plus`, `igd`, `gd`, `spread` against a reference front. @@ -1313,4 +1314,4 @@ every = 50 - **Reproducible:** a seed gives the same results on every platform and thread count, parallel or not. The exception is a fitness function that calls the platform's `sin`, `cos`, `exp` and the like (`f64::sin`, numpy): their last bit can differ between operating systems, and long runs drift apart. `genoxide::math::{sin, cos, tan, exp, ln, powf, powi, atan2, ...}` are the same to the bit everywhere, at native speed; `problems` and `multi::problems` use them. - **Ties:** the earlier individual wins. - **The best is kept:** `outcome.best()` is the best individual ever evaluated. -- **Errors, not panics,** for invalid settings, including sizes above 2^24. The only panics (`# Panics`): an index out of bounds (`Bits::set`, `Order::swap`), a `problems` or `multi::problems` constructor with too few dimensions or variables (or a radius that isn't above 0, or none from the paper for `C1Dtlz3::new` and `ConvexC2Dtlz2::new`), a `Batch` returning no score for a single genome, a gradient or constraint-values slice of the wrong length for a test problem's `evaluate_with`, an input, output or observation slice of the wrong length for an `nn` network or a `control` task, and a point of the wrong length for a `model::gp::GaussianProcess`'s predictions or `Bo::acquisition_at`. +- **Errors, not panics,** for invalid settings, including sizes above 2^24. The only panics (`# Panics`): an index out of bounds (`Bits::set`, `Order::swap`), a `problems` or `multi::problems` constructor with too few dimensions or variables (or a radius that isn't above 0, or none from the paper for `C1Dtlz3::new` and `ConvexC2Dtlz2::new`, a DC-DTLZ `a` that isn't above 0 or `b` outside (−1, 1), or a `Difficulty` level outside [0, 1]), a `Batch` returning no score for a single genome, a gradient or constraint-values slice of the wrong length for a test problem's `evaluate_with`, an input, output or observation slice of the wrong length for an `nn` network or a `control` task, and a point of the wrong length for a `model::gp::GaussianProcess`'s predictions or `Bo::acquisition_at`. diff --git a/docs/features.md b/docs/features.md index 1ac765bc..edcca1d5 100644 --- a/docs/features.md +++ b/docs/features.md @@ -84,6 +84,7 @@ What genoxide has on main; [docs.rs](https://docs.rs/genoxide) documents the lat - **Test problems** (`multi::problems`): ZDT1-6 (ZDT5 on bit strings), DTLZ1-7, WFG1-9 (any number of objectives, checked against the authors' toolkit), Schaffer's two, Fonseca and Fleming's, Kursawe's, Poloni's and Viennet's three, and the constrained BNH, SRN, TNK, OSY and CONSTR, each with its optimal front where it's known and its reference, in Rust and Python. - **Constrained test problems of tunable difficulty** (`multi::problems`): CTP1-8 (Deb, Pratap and Meyarivan), whose constraints make the front disconnected, a set of points or hidden behind infeasible bands, and Jain and Deb's constrained DTLZ problems C1-DTLZ1, C1-DTLZ3, C2-DTLZ2, convex C2-DTLZ2, C3-DTLZ1 and C3-DTLZ4 for any number of objectives, with their fronts, in Rust and Python. - **More test problems** (`multi::problems`): Deb and Jain's convex DTLZ2, scaled DTLZ1 and DTLZ2 and inverted DTLZ1, and Ma and Wang's constrained MW1-14, with fronts derived from their definitions, in Rust and Python. +- **Advanced constrained suites** (`multi::problems`): the DTLZ report's constrained DTLZ8 and DTLZ9, Li et al.'s DC-DTLZ problems (DC1-DC3 on DTLZ1 and DTLZ3, constraints on the decision variables) and Fan et al.'s DAS-CMOP1-9, whose difficulty triplet (the paper's sixteen, or any) sets how hard their constraints make diversity, feasibility and convergence, with fronts derived or sampled from their definitions, in Rust and Python. - **Engineering design problems with several objectives** (`multi::problems::engineering`): the two-bar and four-bar trusses, the welded beam, the disc brake and the speed reducer (two objectives), the car side impact, the rocket injector, vehicle crashworthiness and conceptual marine design (three) and water resource planning (five), the trusses' and water resource planning's fronts derived from their definitions, in Rust and Python. ## Engine diff --git a/docs/problems-plan.md b/docs/problems-plan.md index df2b412b..12dec314 100644 --- a/docs/problems-plan.md +++ b/docs/problems-plan.md @@ -460,6 +460,18 @@ All x ∈ [0, 1]ⁿ, n = M + k − 1. | DTLZ8 (eq. 28), constrained | n = 10M | a line (f₁ = … = f_{M−1} = t, f_M = 1 − 4t, t ∈ [0, 1/6]) and a hyperplane 2f_M + fᵢ + fⱼ = 1, f_M ∈ [0, 1/3] (derived) | M constraints; **index erratum:** the block sums start at ⌊(j − 1)n/M⌋ + 1 (1-based) | | DTLZ9 (eq. 29), constrained | n = 10M | f₁ = … = f_{M−1}, f_M² + fⱼ² = 1 | M − 1 constraints | +**Checked in batch 11** (DTLZ8 and DTLZ9, in the report, section 8.8-8.9, eqs. 28-29; no code of +the authors' found). The index erratum is read as above. DTLZ8 needs M ≥ 3 (its last constraint +takes the least sum of two different objectives other than f_M), and its surface part is not the +whole plane but, derived and checked against random feasible points for M = 3 and 4: f_M = t in +[0, 1/3], one of the first M − 1 objectives s in [(1 − t)/4, (1 − 2t)/2] and the others equal, +u = 1 − 2t − s (a point with two unequal larger objectives is dominated by the lesser); ideal 0, +nadir (3/4, …, 3/4, 1). DTLZ9 prints no mean (its objectives are sums, up to ⌈n/M⌉). Examples: +NSGA-II for the report's 500 generations fails on both (DTLZ9: IGD+ 3.4 to 4.8), SMS-EMOA for +20,000 generations reaches 99.5% to 100.7% (DTLZ8) and 99.8% to 99.9% (DTLZ9, IGD+ 0.0025, the +sample's own) of a population-sized sample's hypervolume: the weakly dominated surfaces next to the line +and the curve are what NSGA-II keeps and hypervolume selection drops. + - **Erratum (DTLZ1):** the report's text says the Pareto set is x_M = 0; it is x_M = 0.5 (the CEC paper's eq. 8). genoxide's DTLZ1 uses 0.5 already. - **Numbering:** the CEC 2002 paper has seven problems: its DTLZ5 is the report's DTLZ6, its @@ -717,6 +729,30 @@ v3 (the journal text not compared; v1 had different problems, "DAC-MOP"). ellipse parameters a_k² = 0.3, b_k² = 1.2 (Table 2) vs 0.4 and 1.6 (eq. 5's example), and implementations in circulation that read 0.3 and 1.2 as the semi-axes; d at ζ = 0. +**Checked in batch 11.** Read: the arXiv v3 text (tables 2-5, figure 6), and the authors' Java +code and sampled fronts (their laboratory's page, +: no license, compared +only). Settled by the code, which agrees with the paper otherwise: g sums from j = 2; a_k² = 0.3 +and b_k² = 1.2 are squares; at ζ = 0, d = 0 and e = 10³⁰ (genoxide: the constraint −g); at +ζ = 1, |g − 0.5| ≤ 10⁻⁴. Fronts: the first feasible point of each ray α + g (1, …, 1) (each +ellipse or sphere a quadratic in g along it), non-dominated: 20,000 rays plus the type I +intervals' ends for two objectives, Das and Dennis's 288 divisions for three, cached per triplet. +Against the authors' 144 sampled fronts, the mean distance is under 0.01 both ways for 130 (theirs +to genoxide's) and 140 (genoxide's to theirs); the rest are their points that break the +constraints (g ≠ 0.5 at ζ = 1, x₁ outside the type I intervals), the isolated x₁ = 1 at η = 0.5 +(feasible only in exact arithmetic), and genoxide's points pushed out by the type III spheres near +the corners of DAS-CMOP8/9's front, which their files leave out. Examples (the paper's figure 6 +triplets, N = 300, 300,000 evaluations): NSGA-II with constrained dominance reproduces the +paper's NSGA-II-CDP (DAS-CMOP1 IGD ≈ 0.7, DAS-CMOP2-3 ≈ 0.15); DAS-CMOP1-3 and DAS-CMOP9, with +linked distance variables, take MOEA/D-DE (differential evolution moves the linked variables +together; SBX and polynomial mutation move them one at a time), with parents from the +neighbourhood with probability δ = 0.2: DAS-CMOP1-3 reached in 20 of 20 seeds (δ = 0.9, the +paper's, 18, 13 and 1 of 20, as the violation-only phase gathers the population at one stretch of +x₁), DAS-CMOP9 on the front (median distance about 0.002) at 97.3-98.0% of a sample's +hypervolume, above the 96.7% of the 300 Tchebycheff weights' own best points of the front; +DAS-CMOP4-6 are reached by NSGA-II with polynomial mutation η = 5 (19-20 of 20 seeds), and +DAS-CMOP7-8 by NSGA-III with η = 5 (20 of 20 at 99% of a sample's hypervolume). + #### LIR-CMOP1-14 Fan, Z., Li, W., Cai, X., Huang, H., Fang, Y., You, Y., Mo, J., Wei, C. and Goodman, E. (2019). @@ -752,6 +788,22 @@ DTLZ1 and DTLZ3. **U:** n (M + 4 and M + 9 assumed), DC2's and DC3's a and b val are printed strict (">"). The supplement gives C2-DTLZ2's r as 0.1, which conflicts with Jain and Deb (0.4 / 0.5): genoxide follows Jain and Deb. +**Checked in batch 11.** Read: the supplement and the arXiv paper, and the C++ code of the +authors' laboratory, EMOC (, `src/problem/dcdtlz`, and +its sampled fronts `pf_data/dc*dtlz`; no license, compared only). Settled: a = 3 throughout, +b = 0.5 but for DC2-DTLZ1's 0.9 (EMOC; the supplement gives values for DC1 only); DC3 constrains +the M − 1 position variables and g, M constraints (EMOC and the supplement's table 2, whose +feasible shares, 3⁻ᵐ, fit only that; the supplement's "j = 1, …, m" would make the front +infeasible); DTLZ3's g keeps DTLZ3's factor 100 (EMOC has 10, in its C1-DTLZ3 too); n = M + 4 and +M + 9 stay assumed; the strict ">" is read as ≥. Table 2 gives DC1 a feasible share of 10.1%, +arccos(b)/π for b = 0.95, where the text and EMOC have b = 0.5 (a third): genoxide follows the +text. Fronts derived (DTLZ's front where the position variables are feasible; DC2's whole), +agreeing with EMOC's samples (mean distances at most 0.011 both ways). Examples (NSGA-III with +C-NSGA-III's settings, 1,000 generations on DTLZ1, 2,000 on DTLZ3, 4,000 on DC1-DTLZ3): constrained +dominance reaches DC1 (DC1-DTLZ3 in 20 of 20 runs; 13 of 20 after 2,000 generations) and stalls on DC2 (at the violation's local minima, no feasible solution in +20/20) and DC3 (in a band of g), as the paper's C-NSGA-III does; solving without the constraints +(DC2) or without the one on g (DC3) reaches the front. + ### 1.5 Engineering design None of the paywalled originals (Ragsdell and Phillips, Sandgren, Kannan and Kramer, Golinski, @@ -1194,7 +1246,7 @@ page); a comparison belongs on the problems' own pages. | 9 | done ([#357](https://github.com/tachsin/genoxide/pull/357); the marine design after it, once its original was read) | Engineering design, several objectives (`multi::problems::engineering`, section 1.5's "Checked in batch 9") | Two-bar truss, welded beam (2 objectives), disc brake, car side impact (3 objectives), speed reducer (2 objectives), four-bar truss, water resource planning, rocket injector, vehicle crashworthiness, conceptual marine design (10) | an example per problem: `two_bar_truss`, `welded_beam_2obj`, `disc_brake`, `speed_reducer_2obj`, `four_bar_truss` (NSGA-II), `car_side_impact_3obj` (NSGA-III, Jain and Deb's settings), `rocket_injector`, `vehicle_crashworthiness`, `marine_design` (SMS-EMOA), `water_resource_planning` (SPEA2) | | 10a | done | Remaining low-dimensional and classic scalable functions (section 1.1's "Checked in batch 10a") | Beale, Booth, Matyas, Bohachevsky 1-3, Three-hump camel, Dixon-Price, Trid, Powell, Langermann, Shekel's foxholes, Kowalik, Schwefel 2.21, Schwefel 2.22 (15) | an example per function: `beale`, `booth`, `matyas`, `bohachevsky1` to `bohachevsky3`, `three_hump_camel`, `langermann`, `shekel_foxholes` and `kowalik` (30 seeds each of CMA-ES with and without IPOP, DE, PSO or a GA), `dixon_price` (CMA-ES with IPOP, DE and PSO in 5 and 10 dimensions), and `schwefel_2_21`, `schwefel_2_22`, `trid` and `powell` (CMA-ES, sep-CMA-ES, DE, PSO and a GA to errors of 1 … 1e-8) | | 10b | done | CEC and BBOB-style functions, and the shift / rotation wrappers (section 1.1's "Checked in batch 10b") | `Shifted

`, `Rotated

`; Sum of different powers, Step, Quartic (deterministic: without noise, or with noise seeded from the genome, since fitness functions must be deterministic), Penalized 1 and 2, High-conditioned elliptic, Bent cigar, Discus, Büche-Rastrigin, Non-continuous Rastrigin, Weierstrass, Katsuura, HappyCat, HGBat, Schaffer F7, Rotated hyper-ellipsoid, BBOB different powers (17; the shifted and rotated Rastrigin of CEC 2005 and BBOB are the wrappers around `Rastrigin`) | an example per function: `sum_of_different_powers`, `step`, `quartic` (with and without noise), `rotated_hyper_ellipsoid`, `high_conditioned_elliptic`, `bent_cigar`, `discus` and `different_powers` (CMA-ES, sep-CMA-ES, DE, PSO and a GA to errors of 1 … 1e-8, as they are and shifted and rotated, or rotated), and `schaffer_f7`, `penalized1`, `penalized2`, `buche_rastrigin`, `non_continuous_rastrigin`, `weierstrass`, `katsuura`, `happy_cat` and `hg_bat` (10 seeds each of CMA-ES with and without IPOP, DE, PSO and a GA) | -| 11 | | Advanced constrained multi-objective suites | DAS-CMOP1-9 (with the 16 difficulty triplets as a parameter), DC-DTLZ (DC1-DC3 on DTLZ1/DTLZ3), DTLZ8, DTLZ9 (≈15) | an example per problem | +| 11 | done | Advanced constrained multi-objective suites (sections 1.3's and 1.4's "Checked in batch 11") | DAS-CMOP1-9 (with the 16 difficulty triplets as a parameter), DC-DTLZ (DC1-DC3 on DTLZ1/DTLZ3), DTLZ8, DTLZ9 (17) | an example per problem: `das_cmop1` to `das_cmop3` (MOEA/D-DE, and NSGA-II with the paper's settings), `das_cmop4` to `das_cmop6` (NSGA-II, the paper's settings and η = 5), `das_cmop7`, `das_cmop8` (NSGA-II and NSGA-III), `das_cmop9` (MOEA/D-DE, at its 300 weights' limit, and NSGA-II), `dc1_dtlz1_3obj`, `dc1_dtlz3_3obj` (NSGA-III and SMS-EMOA), `dc2_dtlz1_3obj`, `dc2_dtlz3_3obj`, `dc3_dtlz1_3obj`, `dc3_dtlz3_3obj` (NSGA-III with constrained dominance, and without the constraints, or the one on g), `dtlz8_3obj`, `dtlz9_3obj` (NSGA-II for the report's 500 generations, and SMS-EMOA) | | 12 | | Binary and combinatorial problems | OneMax, LeadingOnes, deceptive trap, royal road, NK landscapes (seeded), 0/1 knapsack (generated instance classes) (6) | an example per problem; `one_max` and `knapsack` switch to the problems | | 13 (optional) | | Competition suites whose definitions are long | LIR-CMOP1-14, CEC 2009 UF1-UF10 and CF1-CF10, MaF1-MaF15, Deb's 1999 two-objective problems, Van Veldhuizen's constrained problems; the deferred engineering problems (section 1.5) once their originals are read | none; used by the benchmark suite | diff --git a/examples/README.md b/examples/README.md index 8bd7dee5..a1746c8e 100644 --- a/examples/README.md +++ b/examples/README.md @@ -176,6 +176,23 @@ python examples/tsp_berlin52/main.py | [Convex C2-DTLZ2 with 3 objectives](convex_c2_dtlz2_3obj/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/convex-c2-dtlz2-3obj) | | [C3-DTLZ1 with 3 objectives](c3_dtlz1_3obj/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/c3-dtlz1-3obj) | | [C3-DTLZ4 with 3 objectives](c3_dtlz4_3obj/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/c3-dtlz4-3obj) | +| [DTLZ8 with 3 objectives](dtlz8_3obj/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/dtlz8-3obj) | +| [DTLZ9 with 3 objectives](dtlz9_3obj/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/dtlz9-3obj) | +| [DC1-DTLZ1 with 3 objectives](dc1_dtlz1_3obj/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/dc1-dtlz1-3obj) | +| [DC1-DTLZ3 with 3 objectives](dc1_dtlz3_3obj/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/dc1-dtlz3-3obj) | +| [DC2-DTLZ1 with 3 objectives](dc2_dtlz1_3obj/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/dc2-dtlz1-3obj) | +| [DC2-DTLZ3 with 3 objectives](dc2_dtlz3_3obj/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/dc2-dtlz3-3obj) | +| [DC3-DTLZ1 with 3 objectives](dc3_dtlz1_3obj/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/dc3-dtlz1-3obj) | +| [DC3-DTLZ3 with 3 objectives](dc3_dtlz3_3obj/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/dc3-dtlz3-3obj) | +| [DAS-CMOP1](das_cmop1/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/das-cmop1) | +| [DAS-CMOP2](das_cmop2/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/das-cmop2) | +| [DAS-CMOP3](das_cmop3/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/das-cmop3) | +| [DAS-CMOP4](das_cmop4/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/das-cmop4) | +| [DAS-CMOP5](das_cmop5/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/das-cmop5) | +| [DAS-CMOP6](das_cmop6/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/das-cmop6) | +| [DAS-CMOP7](das_cmop7/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/das-cmop7) | +| [DAS-CMOP8](das_cmop8/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/das-cmop8) | +| [DAS-CMOP9](das_cmop9/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/das-cmop9) | | [Convex DTLZ2 with 3 objectives](convex_dtlz2/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/convex-dtlz2) | | [Scaled DTLZ1 with 3 objectives](scaled_dtlz1/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/scaled-dtlz1) | | [Scaled DTLZ2 with 3 objectives](scaled_dtlz2/) | multi-objective | Rust, Python | [tachsin.gr](https://tachsin.gr/projects/genoxide/examples/scaled-dtlz2) | diff --git a/examples/car_side_impact_3obj/README.md b/examples/car_side_impact_3obj/README.md index 973ac487..842f4afa 100644 --- a/examples/car_side_impact_3obj/README.md +++ b/examples/car_side_impact_3obj/README.md @@ -6,7 +6,7 @@ reference: "Jain, H. and Deb, K. (2014). An evolutionary many-objective optimiza reference_url: https://doi.org/10.1109/TEVC.2013.2281534 optimum: "not known in closed form; ideal point (23.585658, 3.58525, 10.610644); genoxide's reference front has a hypervolume of 0.8687 in objectives scaled by the ideal point and its estimated nadir point (42.768, 4.0, 12.5212) (reference point (1.1, 1.1, 1.1))" languages: [rust, python] -order: 225 +order: 241 --- # Car side impact, three objectives diff --git a/examples/convex_dtlz2/README.md b/examples/convex_dtlz2/README.md index 8dddba7c..06f0cbf4 100644 --- a/examples/convex_dtlz2/README.md +++ b/examples/convex_dtlz2/README.md @@ -6,7 +6,7 @@ reference: "Deb, K. and Jain, H. (2014). An evolutionary many-objective optimiza reference_url: https://doi.org/10.1109/TEVC.2013.2281535 optimum: "the surface f₃ + √f₁ + √f₂ = 1 with fᵢ in [0, 1]; hypervolume 1.2943 (reference point (1.1, 1.1, 1.1))" languages: [rust, python] -order: 200 +order: 216 family: DTLZ tab: Convex DTLZ2 --- diff --git a/examples/das_cmop1/README.md b/examples/das_cmop1/README.md new file mode 100644 index 00000000..71fb208c --- /dev/null +++ b/examples/das_cmop1/README.md @@ -0,0 +1,74 @@ +--- +title: DAS-CMOP1 +category: multi-objective +summary: Minimize two objectives over 30 variables subject to 11 constraints of adjustable difficulty, whose Pareto front is pieces of a concave curve, with MOEA/D-DE, which reaches it in every run of 20, and NSGA-II, which doesn't, as the paper's doesn't. +reference: "Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). Difficulty adjustable and scalable constrained multiobjective test problem toolkit. Evolutionary Computation 28(3): 339-378." +reference_url: https://doi.org/10.1162/evco_a_00259 +optimum: "for the difficulty triplet (0, 0.5, 0.5), two pieces of the curve f₂ = 1.5 − (f₁ − 0.5)²; ideal point (0.5, 0.5), nadir point (1.5, 1.5); hypervolume 0.4706 (normalized objectives, reference point (1.1, 1.1))" +languages: [rust, python] +order: 207 +family: DAS-CMOP +--- + +# DAS-CMOP1 + +## The problem + +Fan et al. (2020) built a toolkit of constrained test problems whose difficulty is set by three numbers, a triplet (η, ζ, γ) in [0, 1]³, one for each kind of difficulty: η for diversity, ζ for feasibility and γ for convergence. Each kind comes from a type of constraint. Type I constraints cut the front into pieces, narrower as η grows. A type II constraint asks the distance from the unconstrained front to be in a band, narrower as ζ grows. Type III constraints are infeasible regions in the objective space that block the way to the front, larger as γ grows. The paper suggests nine problems built this way, DAS-CMOP1 to DAS-CMOP9, and runs each with sixteen triplets (its table 3). + +DAS-CMOP1 (table 2) has two objectives over 30 variables in [0, 1]: + +```text +f₁ = x₁ + g +f₂ = 1 − x₁² + g +g = Σⱼ₌₂³⁰ (xⱼ − sin(0.5πx₁))² +subject to + sin(20πx₁) − b ≥ 0, b = 2η − 1 (type I) + (e − g)(g − 0.5) ≥ 0, e = 0.5 − ln ζ (type II) + ((f₁ − p_k) cos θ − (f₂ − q_k) sin θ)²/0.3 + + ((f₁ − p_k) sin θ + (f₂ − q_k) cos θ)²/1.2 ≥ r, r = γ/2 (type III, k = 1, …, 9) +θ = −π/4, (p_k, q_k) = (0, 1.5), (1, 0.5), (0, 2.5), (1, 1.5), (2, 0.5), (0, 3.5), (1, 2.5), (2, 1.5), (3, 0.5) +``` + +Both objectives are minimized. At g = 0 the unconstrained front is concave. The distance function g is 0 where every xⱼ equals sin(0.5πx₁): the variables are linked, each other variable's best value depending on x₁. The example uses the triplet with which the paper's figure 6 plots DAS-CMOP1, (0, 0.5, 0.5). Then the type I constraint has no effect (b = −1); g must be between 0.5 and 0.5 + ln 2 ≈ 1.193, which moves the front out along the diagonal by 0.5; and the nine ellipses, each 45° from the axes, have semi-axes √(0.3 r) ≈ 0.27 along the diagonal and √(1.2 r) ≈ 0.55 across it. + +The front depends on the triplet, and the paper samples it rather than writing it down; genoxide samples it in the same way, as the first feasible point of each of 20,000 rays α(x₁) + g (1, 1) that the type I constraint allows, non-dominated. With this triplet it is two pieces of the shifted curve: f₁ from 0.5 to 0.6722 and from 1.1424 to 1.5. Between them, the curve runs through the ellipse centered at (1, 1.5), and the points it pushes out are dominated. Its ideal point is (0.5, 0.5) and its nadir point (1.5, 1.5). The authors published sampled fronts for the sixteen triplets with their code, which genoxide's agree with (the docs of `multi::problems::DasCmop1` say how they were compared). + +## What makes it hard + +The linked variables first. A solution on the front at x₁ has every other variable at sin(0.5πx₁): to move along the front, all 29 of them have to move together, by the same amount. Simulated binary crossover and polynomial mutation change each variable on its own, so a population that has settled on one value of x₁ stays there: any other x₁ costs 29 (Δ sin)² in g. Differential evolution moves them together: its steps are differences between solutions of the population, along the front once the population lies near it. The paper's MOEA/D-CDP uses it, and so does MOEA/D-DE here. Then the constraints: the band of g between 0.5 and e is feasible, a thin shell above the front, and the ellipses block parts of it. + +## Representation + +A `Real` genome of 30 genes in [0, 1]. The problem is genoxide's `DasCmop1`, whose fitness is the two objectives and the total constraint violation, 0 when it is feasible; the triplet is a `Difficulty`, `Difficulty::standard(k)` for the paper's sixteen. In Python, `gx.problems.DasCmop1()`, with `difficulty=(η, ζ, γ)` or the number of one of the sixteen, which `run` evaluates in Rust, so both versions print the same. + +Solutions compare by constrained dominance, the rule of the NSGA-II paper and the constraint handling that the paper calls CDP. A feasible solution beats an infeasible one. Of two infeasible ones, the smaller violation wins. Of two feasible ones, Pareto dominance decides. + +## Algorithm + +Two runs, each with the paper's population of 300 for 1,000 generations, 300,000 evaluations (section 7.1), and polynomial mutation with η = 20 at a rate of 1/30 per gene: + +- MOEA/D with differential evolution, MOEA/D-DE (Li and Zhang, 2009, IEEE Transactions on Evolutionary Computation 13(2): 284-302): a subproblem for each of 300 weight vectors spread evenly on the line w₁ + w₂ = 1, each the weighted Tchebycheff distance to the ideal point, with the paper's 30 neighbours (0.1 N) and at most 2 replacements per child; a child is its subproblem's solution moved by F = 0.5 times the difference of two parents, every gene (CR = 1), the parents from the neighbourhood with probability δ = 0.2; +- NSGA-II (Deb, Pratap, Agarwal and Meyarivan, 2002, IEEE Transactions on Evolutionary Computation 6(2): 182-197) with the paper's settings, simulated binary crossover with η = 20 at a rate of 0.9, as the contrast. + +Both compare solutions by constrained dominance. Differential evolution moves the linked variables together, which simulated binary crossover can't. + +δ = 0.2, where the paper and Li and Zhang use 0.9, keeps the population spread. While no solution is feasible, constrained dominance ranks solutions by their violation alone, and that pulls the whole population to one stretch of x₁: with seed 1 and δ = 0.9, 80% of the population has x₁ between 0.27 and 0.41 after 10 generations. Parents from the whole population, most of the time, spread it again once it is feasible; parents from the neighbourhood, nine times in ten, mostly can't, as the neighbourhood's solutions are all at that x₁. Over seeds 1 to 20, δ = 0.9 reaches the target in 18 runs of 20 (IGD+ from 0.0008 to 0.0015 and 98.4% to 99.8% of the sample's hypervolume). + +## Output + +A line per run: the size of its final front, how many of its solutions are feasible, the front's IGD+ and hypervolume, and the hypervolume as a share of that of a sample of 300 points of the optimal front, as many as the population. Then the hypervolumes of the whole optimal front and of the sample. + +Both indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 500 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest point of the found front, counting only the objectives in which the found point is worse. Smaller is better, and 0 means the found front covers the optimal one. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the area the front dominates up to the reference point (1.1, 1.1). Larger is better; the whole front's is computed from 20,000 of its points, and the sample's is what a front of 300 points reaches. Only feasible solutions count. + +The page plays both runs back over the grey feasible region, sampled from genomes with x₁ evenly spread and every distance variable at one value, swept; hollow points are infeasible solutions of the populations (at most 100 a frame), and the line is the optimal front, in its pieces. + +[The project page](https://tachsin.gr/projects/genoxide/examples/das-cmop1) plays this run back. + +## Good results + +The target: every solution feasible, and 99% of the hypervolume of the sample of 300 points of the front, about what a front of 300 solutions spread like it has. + +MOEA/D-DE's run with seed 1 ends with 164 solutions, 164 feasible, IGD+ 0.0010, hypervolume 0.4681, 99.6% of the sample's; NSGA-II's with 218 solutions, 218 feasible, IGD+ 0.6811, hypervolume 0.0214, 4.5% of the sample's. Over seeds 1 to 20, MOEA/D-DE ends with IGD+ from 0.0008 to 0.0010 and 99.3% to 99.8% of the sample's hypervolume, every run reaching the target, and NSGA-II with IGD+ from 0.6638 to 0.7556 and 0.0% to 6.1% of the sample's hypervolume, no run reaching it: NSGA-II converges to a short stretch of the front, or of the band above it, and stays there, for the reasons above. + +On DAS-CMOP1 with the paper's triplets, NSGA-II-CDP ends with a mean IGD from 0.28 to 0.71 (its table 4; 0.370 with the first triplet, (0.25, 0, 0)), where MOEA/D-CDP, with differential evolution, has 0.0013 with that triplet. diff --git a/examples/das_cmop1/main.py b/examples/das_cmop1/main.py new file mode 100644 index 00000000..778051a4 --- /dev/null +++ b/examples/das_cmop1/main.py @@ -0,0 +1,101 @@ +"""DAS-CMOP1: minimize two objectives over 30 variables subject to 11 constraints, whose front is +pieces of a concave curve, with MOEA/D-DE and NSGA-II. + +Fan et al.'s DAS-CMOP1, from genoxide's problems.DasCmop1, with the difficulty triplet of the paper's +figure 6, (0, 0.5, 0.5); run evaluates it in Rust. Runs MOEA/D with differential evolution +(MOEA/D-DE) and, as a contrast, NSGA-II with the paper's settings, a population of 300 for 1,000 +generations each (300,000 evaluations), with constraint dominance. Prints each final front's size, +how many of its solutions are feasible, and its IGD+ to 500 points of the optimal front and +hypervolume, with the objectives normalized by the front's ideal and nadir points, as a share of +that of a sample of the front with as many points as the population; then the hypervolumes of the +whole front and of the sample. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/das_cmop1/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points: 1.1 times the nadir point +REFERENCE = [1.1, 1.1] + +# the paper's population and evaluations: 300 for 1,000 generations +POPULATION = 300 +GENERATIONS = 1_000 + +problem = gx.problems.DasCmop1() +ideal, nadir = problem.ideal_point, problem.nadir_point +rate = 1 / problem.dimensions + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) +# the sample: as many points of the optimal front as the population has, about what a front of that +# many solutions can be +sample = normalized(problem.optimal_front(POPULATION)) + + +def run(name, algorithm): + """Runs ``algorithm`` for the paper's 1,000 generations, and prints its final front's size, how + many of it are feasible, and its IGD+ to 500 points of the optimal front and hypervolume, with + normalized objectives, as a share of the sample's.""" + result = algorithm.run(problem, generations=GENERATIONS, on_generation=trace.fronts(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"{name}, {GENERATIONS} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(500)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + percent = 100 * volume / gx.indicators.hypervolume(sample, REFERENCE) + print( + start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}, " + f"{percent:.1f}% of the sample's" + ) + + +# MOEA/D-DE: a subproblem per weight vector, 30 neighbours (the paper's 0.1 N), parents from the +# neighbourhood with probability 0.2, differential evolution with F = 0.5 and CR = 1 +moead = gx.Moead( + problem.genome, + objectives=problem.objectives, + weights=gx.das_dennis(2, POPULATION - 1), + neighbors=30, + neighbor_mating=0.2, + crossover=gx.DifferentialEvolutionCrossover(f=0.5, cr=1.0), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("MOEA/D-DE", moead) +# the contrast: NSGA-II with the paper's settings +nsga2 = gx.Nsga2( + problem.genome, + objectives=problem.objectives, + population_size=POPULATION, + crossover=gx.SimulatedBinaryCrossover(20), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-II", nsga2) +# the hypervolume of the whole front, from 20,000 of its points +whole = normalized(problem.optimal_front(20_000)) +print( + f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}; " + f"a sample of {len(sample)} of its points: " + f"{gx.indicators.hypervolume(sample, REFERENCE):.4f}" +) +trace.write() diff --git a/examples/das_cmop1/main.rs b/examples/das_cmop1/main.rs new file mode 100644 index 00000000..cc88dd5e --- /dev/null +++ b/examples/das_cmop1/main.rs @@ -0,0 +1,135 @@ +//! DAS-CMOP1: minimize two objectives over 30 variables subject to 11 constraints, whose front is +//! pieces of a concave curve, with MOEA/D-DE and NSGA-II. +//! +//! Fan et al.'s DAS-CMOP1, from genoxide's `multi::problems::DasCmop1`, with the difficulty triplet of +//! the paper's figure 6, (0, 0.5, 0.5). Runs MOEA/D with differential evolution (MOEA/D-DE) and, +//! as a contrast, NSGA-II with the paper's settings, a population of 300 for 1,000 generations +//! each (300,000 evaluations), with constraint dominance. Prints each final front's size, how many +//! of its solutions are feasible, and its IGD+ to 500 points of the optimal front and hypervolume, +//! with the objectives normalized by the front's ideal and nadir points, as a share of that of a +//! sample of the front with as many points as the population; then the hypervolumes of the whole +//! front and of the sample. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example das_cmop1 +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{DasCmop1, MultiProblem}; +use genoxide::multi::{DifferentialEvolutionCrossover, MultiFitnessFunction}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points: 1.1 times the nadir point +const REFERENCE: [f64; 2] = [1.1, 1.1]; + +// the paper's population and evaluations: 300 for 1,000 generations +const POPULATION: usize = 300; +const GENERATIONS: u64 = 1_000; + +fn main() -> Result<()> { + let problem = DasCmop1::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the sample: as many points of the optimal front as the population has, about what a front + // of that many solutions can be + let sample = normalized(&problem, &problem.optimal_front(POPULATION).expect("known")); + let rate = 1.0 / problem.variables() as f64; + // MOEA/D-DE: a subproblem per weight vector, 30 neighbours (the paper's 0.1 N), parents from + // the neighbourhood with probability 0.2, differential evolution with F = 0.5 and CR = 1 + let moead = Moead::builder( + problem.representation(), + [Minimize; 2], + multi::das_dennis::<2>(POPULATION - 1), + ) + .neighbors(30) + .neighbor_mating(0.2) + .crossover(DifferentialEvolutionCrossover::new(0.5, 1.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run(&problem, "MOEA/D-DE", moead, &sample, &mut trace)?; + // the contrast: NSGA-II with the paper's settings + let nsga2 = Nsga2::builder(problem.representation(), [Minimize; 2]) + .population_size(POPULATION) + .crossover(SimulatedBinaryCrossover::new(20.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run(&problem, "NSGA-II", nsga2, &sample, &mut trace)?; + // the hypervolume of the whole front, from 20,000 of its points + let whole = normalized(&problem, &problem.optimal_front(20_000).expect("known")); + println!( + "the whole front: hypervolume {:.4}; a sample of {} of its points: {:.4}", + hypervolume(&whole, &REFERENCE, &[Minimize; 2]), + sample.len(), + hypervolume(&sample, &REFERENCE, &[Minimize; 2]) + ); + trace.write(&problem); + Ok(()) +} + +// runs `algorithm` for the paper's 1,000 generations, and prints its final front's size, how many +// of it are feasible, and its IGD+ to 500 points of the optimal front and hypervolume, with +// normalized objectives, as a share of the sample's +fn run( + problem: &DasCmop1, + name: &'static str, + algorithm: A, + sample: &[[f64; 2]], + trace: &mut trace::Trace, +) -> Result<()> +where + A: MultiObjectiveAlgorithm<2, Genome = Reals>, +{ + let mut record = trace.fronts(name); + let outcome = MultiEngine::new(algorithm, *problem) + .stop_when(Stop::generations(GENERATIONS)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 2], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 2]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "{name}, {GENERATIONS} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(500).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 2]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 2]); + let percent = 100.0 * volume / hypervolume(sample, &REFERENCE, &[Minimize; 2]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the \ + sample's", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &DasCmop1, points: &[[f64; 2]]) -> Vec<[f64; 2]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 2]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/das_cmop1/output.txt b/examples/das_cmop1/output.txt new file mode 100644 index 00000000..95ecc234 --- /dev/null +++ b/examples/das_cmop1/output.txt @@ -0,0 +1,3 @@ +MOEA/D-DE, 1000 generations: 164 solutions, 164 feasible, IGD+ 0.0010, hypervolume 0.4681, 99.6% of the sample's +NSGA-II, 1000 generations: 218 solutions, 218 feasible, IGD+ 0.6811, hypervolume 0.0214, 4.5% of the sample's +the whole front: hypervolume 0.4706; a sample of 300 of its points: 0.4701 diff --git a/examples/das_cmop1/trace.json b/examples/das_cmop1/trace.json new file mode 100644 index 00000000..17f6eb1f --- /dev/null +++ b/examples/das_cmop1/trace.json @@ -0,0 +1,42 @@ 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+]} diff --git a/examples/das_cmop1/trace.py b/examples/das_cmop1/trace.py new file mode 100644 index 00000000..662dcc2f --- /dev/null +++ b/examples/das_cmop1/trace.py @@ -0,0 +1,260 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: both fronts, each population's infeasible solutions (at most 100) and +feasible share, and the fronts' hypervolumes with normalized objectives, in at most 40 +generations; and the optimal front, in its pieces, and the feasible region, sampled. A frame's +evaluations are the two runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the samples of the feasible region along x₁ and along the distance variables, its grid, and the +# most infeasible solutions a frame shows +SAMPLES = 400 +DISTANCES = 800 +CELLS = 80 +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions, the hypervolume and the feasible share + self.series = {} + + def fronts(self, name): + """The callback for ``run`` that records the run ``name`` after each generation: None + without a trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + feasible = progress.front_violations == 0 + front = progress.front_objectives[feasible] + infeasible = progress.objectives[progress.violations != 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + share = (progress.violations == 0).sum() / len(progress.violations) + history.append( + (progress.evaluations, front.tolist(), infeasible.tolist(), volume, share) + ) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(40) + for generation in range(max(map(len, self.series.values()), default=0)): + at = {name: runs[min(generation, len(runs) - 1)] for name, runs in self.series.items()} + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at.values()), + "best": None, + "median": None, + "state": { + "fronts": {name: record[1] for name, record in at.items()}, + "infeasible": {name: record[2] for name, record in at.items()}, + "hypervolume": {name: record[3] for name, record in at.items()}, + "feasible": {name: record[4] for name, record in at.items()}, + }, + } + ) + whole = self.normalized(self.problem.optimal_front(20_000)) + settings = { + "format": 1, + "example": "das_cmop1", + "objective": ["minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "front-2d", + "problem": { + "objectives": ["f1", "f2"], + "true_front": pieces(self.problem.optimal_front(400).tolist()), + "series": list(self.series), + "feasible_region": region(self.problem), + }, + } + write(self.path, settings, frames.to_list()) + + +def pieces(front): + """The front in pieces, split where two points are more than 2% of its extent apart, each + point once.""" + first, last = front[0], front[-1] + extent = math.hypot(last[0] - first[0], first[1] - last[1]) + split = [] + for point in front: + if not split: + split.append([point]) + continue + previous = split[-1][-1] + if math.hypot(point[0] - previous[0], point[1] - previous[1]) > 0.02 * extent: + split.append([point]) + elif point != previous: + split[-1].append(point) + return split + + +def region(problem): + """The feasible region in objective space: genomes with x₁ evenly spread over [0, 1] and + every distance variable at a value v swept over [0, 1], which moves g from 0 up; a cell of a + grid of 80 × 80 over [0, 1.25 f₁ᵐᵃˣ] × [0, 1.25 f₂ᵐᵃˣ] (the nadir point's) is feasible if a + feasible genome lands in it, and the rows go up from f₂ = 0.""" + n = problem.dimensions + nadir = problem.nadir_point + width, height = 1.25 * nadir[0], 1.25 * nadir[1] + cells = [[False] * CELLS for _ in range(CELLS)] + for a in range(SAMPLES): + genomes = np.empty((DISTANCES, n)) + genomes[:, 0] = a / (SAMPLES - 1) + for b in range(DISTANCES): + genomes[b, 1:] = 0.0 + 1.0 * b / (DISTANCES - 1) + objectives, violations = problem.evaluate(genomes) + i = np.floor(objectives[:, 0] / width * CELLS) + j = np.floor(objectives[:, 1] / height * CELLS) + inside = (violations == 0) & (i >= 0) & (j >= 0) & (i < CELLS) & (j < CELLS) + for column, row in zip(i[inside].astype(int), j[inside].astype(int)): + cells[row][column] = True + # each row as the pairs of columns [start, end) where it's feasible + rows = [runs(row) for row in cells] + return {"x": [0.0, width], "y": [0.0, height], "columns": CELLS, "rows": rows} + + +def runs(row): + """The columns where a row of cells turns feasible or infeasible, as pairs [start, end).""" + turns = [ + column for column in range(len(row)) if row[column] != (column > 0 and row[column - 1]) + ] + return turns + ([len(row)] if row[-1] else []) + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/das_cmop1/trace.rs b/examples/das_cmop1/trace.rs new file mode 100644 index 00000000..35765f7b --- /dev/null +++ b/examples/das_cmop1/trace.rs @@ -0,0 +1,367 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: both fronts, each population's infeasible solutions (at most 100) and +//! feasible share, and the fronts' hypervolumes with normalized objectives, in at most 40 +//! generations; and the optimal front, in its pieces, and the feasible region, sampled. A frame's +//! evaluations are the two runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{DasCmop1, MultiProblem}; +use genoxide::multi::{MultiFitnessFunction, MultiSnapshot, Scores}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, the +// hypervolume and the feasible share +type Record = (u64, Vec<[f64; 2]>, Vec<[f64; 2]>, f64, f64); + +// the samples of the feasible region along x₁ and along the distance variables, its grid, and the +// most infeasible solutions a frame shows +const SAMPLES: usize = 400; +const DISTANCES: usize = 800; +const CELLS: usize = 80; +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation + pub fn fronts(&mut self, name: &'static str) -> impl FnMut(&MultiSnapshot<'_, Reals, 2>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let values = |x: &Individual>| x.fitness()?.values(); + let feasible = |x: &&Individual>| { + x.fitness().is_some_and(|scores| scores.is_feasible()) + }; + let front: Vec<[f64; 2]> = snapshot + .front() + .iter() + .filter(feasible) + .filter_map(values) + .collect(); + let population = snapshot.population().as_slice(); + let infeasible = population.iter().filter(|x| !feasible(x)); + let infeasible: Vec<[f64; 2]> = infeasible.filter_map(values).take(SHOWN).collect(); + let share = population.iter().filter(feasible).count() as f64 / population.len() as f64; + let volume = hypervolume( + &normalized(&DasCmop1::default(), &front), + &REFERENCE, + &[Minimize; 2], + ); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume, share)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &DasCmop1) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(40); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let mut state = [Map::new(), Map::new(), Map::new(), Map::new()]; + let mut evaluations = 0; + for (name, history) in &self.series { + let (done, front, infeasible, volume, share) = + &history[generation.min(history.len() - 1)]; + let values = [json!(front), json!(infeasible), json!(volume), json!(share)]; + for (map, value) in state.iter_mut().zip(values) { + map.insert(name.to_string(), value); + } + evaluations += done; + } + let [fronts, infeasible, volumes, shares] = state; + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "state": { + "fronts": fronts, + "infeasible": infeasible, + "hypervolume": volumes, + "feasible": shares, + }, + })); + } + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(20_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "das_cmop1", + "objective": ["minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 2]), + "plot": "front-2d", + "problem": { + "objectives": ["f1", "f2"], + "true_front": pieces(&problem.optimal_front(400).expect("known")), + "series": names, + "feasible_region": region(problem), + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the front in pieces, split where two points are more than 2% of its extent apart, each point +// once +fn pieces(front: &[[f64; 2]]) -> Vec> { + let (first, last) = (front[0], front[front.len() - 1]); + let extent = (last[0] - first[0]).hypot(first[1] - last[1]); + let mut pieces: Vec> = Vec::new(); + for &point in front { + match pieces.last_mut() { + Some(piece) => { + let previous = piece[piece.len() - 1]; + if (point[0] - previous[0]).hypot(point[1] - previous[1]) > 0.02 * extent { + pieces.push(vec![point]); + } else if point != previous { + piece.push(point); + } + } + None => pieces.push(vec![point]), + } + } + pieces +} + +// the feasible region in objective space: genomes with x₁ evenly spread over [0, 1] and every +// distance variable at a value v swept over [0, 1], which moves g from 0 up; a cell of a grid +// of 80 × 80 over [0, 1.25 f₁ᵐᵃˣ] × [0, 1.25 f₂ᵐᵃˣ] (the nadir point's) is feasible if a feasible +// genome lands in it, and the rows go up from f₂ = 0 +fn region(problem: &DasCmop1) -> Value { + let n = problem.variables(); + let nadir = problem.nadir_point().expect("known"); + let (width, height) = (1.25 * nadir[0], 1.25 * nadir[1]); + let mut cells = vec![vec![false; CELLS]; CELLS]; + for a in 0..SAMPLES { + let mut x = vec![a as f64 / (SAMPLES - 1) as f64; 1]; + x.resize(n, 0.0); + for b in 0..DISTANCES { + let v = 0.0 + 1.0 * b as f64 / (DISTANCES - 1) as f64; + x[1..].fill(v); + let (f, violation) = problem.evaluate(&Reals::from(x.clone())); + let i = (f[0] / width * CELLS as f64).floor(); + let j = (f[1] / height * CELLS as f64).floor(); + if violation == 0.0 && i >= 0.0 && j >= 0.0 && i < CELLS as f64 && j < CELLS as f64 { + cells[j as usize][i as usize] = true; + } + } + } + // each row as the pairs of columns [start, end) where it's feasible + let rows: Vec> = cells.iter().map(Vec::as_slice).map(runs).collect(); + json!({"x": [0.0, width], "y": [0.0, height], "columns": CELLS, "rows": rows}) +} + +// the columns where a row of cells turns feasible or infeasible, as pairs [start, end) +fn runs(row: &[bool]) -> Vec { + let mut runs = Vec::new(); + for (column, &inside) in row.iter().enumerate() { + if inside != (column > 0 && row[column - 1]) { + runs.push(column); + } + } + if row.last() == Some(&true) { + runs.push(row.len()); + } + runs +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/das_cmop2/README.md b/examples/das_cmop2/README.md new file mode 100644 index 00000000..582f97e3 --- /dev/null +++ b/examples/das_cmop2/README.md @@ -0,0 +1,74 @@ +--- +title: DAS-CMOP2 +category: multi-objective +summary: Minimize two objectives over 30 variables subject to 11 constraints of adjustable difficulty, whose Pareto front is pieces of a convex curve and of ellipses, with MOEA/D-DE, which reaches it in every run of 20, and NSGA-II, which doesn't, as the paper's doesn't. +reference: "Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). Difficulty adjustable and scalable constrained multiobjective test problem toolkit. Evolutionary Computation 28(3): 339-378." +reference_url: https://doi.org/10.1162/evco_a_00259 +optimum: "for the difficulty triplet (0, 0.5, 0.5), the curve f₂ = 1.5 − √(f₁ − 0.5) and a piece of an ellipse; ideal point (0.5, 0.5), nadir point (1.5, 1.5); hypervolume 0.8612 (normalized objectives, reference point (1.1, 1.1))" +languages: [rust, python] +order: 208 +family: DAS-CMOP +--- + +# DAS-CMOP2 + +## The problem + +Fan et al. (2020) built a toolkit of constrained test problems whose difficulty is set by three numbers, a triplet (η, ζ, γ) in [0, 1]³, one for each kind of difficulty: η for diversity, ζ for feasibility and γ for convergence. Each kind comes from a type of constraint. Type I constraints cut the front into pieces, narrower as η grows. A type II constraint asks the distance from the unconstrained front to be in a band, narrower as ζ grows. Type III constraints are infeasible regions in the objective space that block the way to the front, larger as γ grows. The paper suggests nine problems built this way, DAS-CMOP1 to DAS-CMOP9, and runs each with sixteen triplets (its table 3). + +DAS-CMOP2 (table 2) has two objectives over 30 variables in [0, 1]: + +```text +f₁ = x₁ + g +f₂ = 1 − √x₁ + g +g = Σⱼ₌₂³⁰ (xⱼ − sin(0.5πx₁))² +subject to + sin(20πx₁) − b ≥ 0, b = 2η − 1 (type I) + (e − g)(g − 0.5) ≥ 0, e = 0.5 − ln ζ (type II) + ((f₁ − p_k) cos θ − (f₂ − q_k) sin θ)²/0.3 + + ((f₁ − p_k) sin θ + (f₂ − q_k) cos θ)²/1.2 ≥ r, r = γ/2 (type III, k = 1, …, 9) +θ = −π/4, (p_k, q_k) = (0, 1.5), (1, 0.5), (0, 2.5), (1, 1.5), (2, 0.5), (0, 3.5), (1, 2.5), (2, 1.5), (3, 0.5) +``` + +Both objectives are minimized. At g = 0 the unconstrained front is convex. The distance function g is 0 where every xⱼ equals sin(0.5πx₁): the variables are linked, each other variable's best value depending on x₁. The example uses the triplet with which the paper's figure 6 plots DAS-CMOP2, (0, 0.5, 0.5). Then the type I constraint has no effect (b = −1); g must be between 0.5 and 0.5 + ln 2 ≈ 1.193, which moves the front out along the diagonal by 0.5; and the nine ellipses, each 45° from the axes, have semi-axes √(0.3 r) ≈ 0.27 along the diagonal and √(1.2 r) ≈ 0.55 across it. + +The front depends on the triplet, and the paper samples it rather than writing it down; genoxide samples it in the same way, as the first feasible point of each of 20,000 rays α(x₁) + g (1, 1) that the type I constraint allows, non-dominated. With this triplet it is one piece from (0.5, 1.5) to (1.5, 0.5): the shifted curve, but for f₁ from 0.839 to 1.248, where the curve runs through the ellipse centered at (1, 0.5) and the front follows that ellipse's boundary, where the rays of the curve's points leave it. Its ideal point is (0.5, 0.5) and its nadir point (1.5, 1.5). The authors published sampled fronts for the sixteen triplets with their code, which genoxide's agree with (the docs of `multi::problems::DasCmop1` say how they were compared). + +## What makes it hard + +The linked variables first. A solution on the front at x₁ has every other variable at sin(0.5πx₁): to move along the front, all 29 of them have to move together, by the same amount. Simulated binary crossover and polynomial mutation change each variable on its own, so a population that has settled on one value of x₁ stays there: any other x₁ costs 29 (Δ sin)² in g. Differential evolution moves them together: its steps are differences between solutions of the population, along the front once the population lies near it. The paper's MOEA/D-CDP uses it, and so does MOEA/D-DE here. Then the constraints: the band of g between 0.5 and e is feasible, a thin shell above the front, and the ellipses block parts of it. + +## Representation + +A `Real` genome of 30 genes in [0, 1]. The problem is genoxide's `DasCmop2`, whose fitness is the two objectives and the total constraint violation, 0 when it is feasible; the triplet is a `Difficulty`, `Difficulty::standard(k)` for the paper's sixteen. In Python, `gx.problems.DasCmop2()`, with `difficulty=(η, ζ, γ)` or the number of one of the sixteen, which `run` evaluates in Rust, so both versions print the same. + +Solutions compare by constrained dominance, the rule of the NSGA-II paper and the constraint handling that the paper calls CDP. A feasible solution beats an infeasible one. Of two infeasible ones, the smaller violation wins. Of two feasible ones, Pareto dominance decides. + +## Algorithm + +Two runs, each with the paper's population of 300 for 1,000 generations, 300,000 evaluations (section 7.1), and polynomial mutation with η = 20 at a rate of 1/30 per gene: + +- MOEA/D with differential evolution, MOEA/D-DE (Li and Zhang, 2009, IEEE Transactions on Evolutionary Computation 13(2): 284-302): a subproblem for each of 300 weight vectors spread evenly on the line w₁ + w₂ = 1, each the weighted Tchebycheff distance to the ideal point, with the paper's 30 neighbours (0.1 N) and at most 2 replacements per child; a child is its subproblem's solution moved by F = 0.5 times the difference of two parents, every gene (CR = 1), the parents from the neighbourhood with probability δ = 0.2; +- NSGA-II (Deb, Pratap, Agarwal and Meyarivan, 2002, IEEE Transactions on Evolutionary Computation 6(2): 182-197) with the paper's settings, simulated binary crossover with η = 20 at a rate of 0.9, as the contrast. + +Both compare solutions by constrained dominance. Differential evolution moves the linked variables together, which simulated binary crossover can't. + +δ = 0.2, where the paper and Li and Zhang use 0.9, keeps the population spread. While no solution is feasible, constrained dominance ranks solutions by their violation alone, and that pulls the whole population to one stretch of x₁: with seed 1 and δ = 0.9, 80% of the population has x₁ between 0.31 and 0.43 after 10 generations. Parents from the whole population, most of the time, spread it again once it is feasible; parents from the neighbourhood, nine times in ten, mostly can't, as the neighbourhood's solutions are all at that x₁. Over seeds 1 to 20, δ = 0.9 reaches the target in 13 runs of 20 (IGD+ from 0.0013 to 0.1224 and 75.2% to 99.9% of the sample's hypervolume). + +## Output + +A line per run: the size of its final front, how many of its solutions are feasible, the front's IGD+ and hypervolume, and the hypervolume as a share of that of a sample of 300 points of the optimal front, as many as the population. Then the hypervolumes of the whole optimal front and of the sample. + +Both indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 500 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest point of the found front, counting only the objectives in which the found point is worse. Smaller is better, and 0 means the found front covers the optimal one. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the area the front dominates up to the reference point (1.1, 1.1). Larger is better; the whole front's is computed from 20,000 of its points, and the sample's is what a front of 300 points reaches. Only feasible solutions count. + +The page plays both runs back over the grey feasible region, sampled from genomes with x₁ evenly spread and every distance variable at one value, swept; hollow points are infeasible solutions of the populations (at most 100 a frame), and the line is the optimal front, in its pieces. + +[The project page](https://tachsin.gr/projects/genoxide/examples/das-cmop2) plays this run back. + +## Good results + +The target: every solution feasible, and 99% of the hypervolume of the sample of 300 points of the front, about what a front of 300 solutions spread like it has. + +MOEA/D-DE's run with seed 1 ends with 266 solutions, 266 feasible, IGD+ 0.0017, hypervolume 0.8577, 99.8% of the sample's; NSGA-II's with 253 solutions, 253 feasible, IGD+ 0.1432, hypervolume 0.6155, 71.6% of the sample's. Over seeds 1 to 20, MOEA/D-DE ends with IGD+ from 0.0016 to 0.0018 and 99.7% to 99.8% of the sample's hypervolume, every run reaching the target, and NSGA-II with IGD+ from 0.1276 to 0.1589 and 68.9% to 74.2% of the sample's hypervolume, no run reaching it: NSGA-II converges to a short stretch of the front, or of the band above it, and stays there, for the reasons above. + +On DAS-CMOP2 the paper's NSGA-II-CDP ends with a mean IGD of 0.335 with the first triplet, (0.25, 0, 0), and 0.216 with (0.5, 0.5, 0.5) (table 4), where MOEA/D-CDP, with differential evolution, has 0.00135 with the first. diff --git a/examples/das_cmop2/main.py b/examples/das_cmop2/main.py new file mode 100644 index 00000000..fbeb59a7 --- /dev/null +++ b/examples/das_cmop2/main.py @@ -0,0 +1,101 @@ +"""DAS-CMOP2: minimize two objectives over 30 variables subject to 11 constraints, whose front is +pieces of a convex curve and of ellipses, with MOEA/D-DE and NSGA-II. + +Fan et al.'s DAS-CMOP2, from genoxide's problems.DasCmop2, with the difficulty triplet of the paper's +figure 6, (0, 0.5, 0.5); run evaluates it in Rust. Runs MOEA/D with differential evolution +(MOEA/D-DE) and, as a contrast, NSGA-II with the paper's settings, a population of 300 for 1,000 +generations each (300,000 evaluations), with constraint dominance. Prints each final front's size, +how many of its solutions are feasible, and its IGD+ to 500 points of the optimal front and +hypervolume, with the objectives normalized by the front's ideal and nadir points, as a share of +that of a sample of the front with as many points as the population; then the hypervolumes of the +whole front and of the sample. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/das_cmop2/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points: 1.1 times the nadir point +REFERENCE = [1.1, 1.1] + +# the paper's population and evaluations: 300 for 1,000 generations +POPULATION = 300 +GENERATIONS = 1_000 + +problem = gx.problems.DasCmop2() +ideal, nadir = problem.ideal_point, problem.nadir_point +rate = 1 / problem.dimensions + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) +# the sample: as many points of the optimal front as the population has, about what a front of that +# many solutions can be +sample = normalized(problem.optimal_front(POPULATION)) + + +def run(name, algorithm): + """Runs ``algorithm`` for the paper's 1,000 generations, and prints its final front's size, how + many of it are feasible, and its IGD+ to 500 points of the optimal front and hypervolume, with + normalized objectives, as a share of the sample's.""" + result = algorithm.run(problem, generations=GENERATIONS, on_generation=trace.fronts(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"{name}, {GENERATIONS} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(500)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + percent = 100 * volume / gx.indicators.hypervolume(sample, REFERENCE) + print( + start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}, " + f"{percent:.1f}% of the sample's" + ) + + +# MOEA/D-DE: a subproblem per weight vector, 30 neighbours (the paper's 0.1 N), parents from the +# neighbourhood with probability 0.2, differential evolution with F = 0.5 and CR = 1 +moead = gx.Moead( + problem.genome, + objectives=problem.objectives, + weights=gx.das_dennis(2, POPULATION - 1), + neighbors=30, + neighbor_mating=0.2, + crossover=gx.DifferentialEvolutionCrossover(f=0.5, cr=1.0), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("MOEA/D-DE", moead) +# the contrast: NSGA-II with the paper's settings +nsga2 = gx.Nsga2( + problem.genome, + objectives=problem.objectives, + population_size=POPULATION, + crossover=gx.SimulatedBinaryCrossover(20), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-II", nsga2) +# the hypervolume of the whole front, from 20,000 of its points +whole = normalized(problem.optimal_front(20_000)) +print( + f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}; " + f"a sample of {len(sample)} of its points: " + f"{gx.indicators.hypervolume(sample, REFERENCE):.4f}" +) +trace.write() diff --git a/examples/das_cmop2/main.rs b/examples/das_cmop2/main.rs new file mode 100644 index 00000000..b36d64d4 --- /dev/null +++ b/examples/das_cmop2/main.rs @@ -0,0 +1,135 @@ +//! DAS-CMOP2: minimize two objectives over 30 variables subject to 11 constraints, whose front is +//! pieces of a convex curve and of ellipses, with MOEA/D-DE and NSGA-II. +//! +//! Fan et al.'s DAS-CMOP2, from genoxide's `multi::problems::DasCmop2`, with the difficulty triplet of +//! the paper's figure 6, (0, 0.5, 0.5). Runs MOEA/D with differential evolution (MOEA/D-DE) and, +//! as a contrast, NSGA-II with the paper's settings, a population of 300 for 1,000 generations +//! each (300,000 evaluations), with constraint dominance. Prints each final front's size, how many +//! of its solutions are feasible, and its IGD+ to 500 points of the optimal front and hypervolume, +//! with the objectives normalized by the front's ideal and nadir points, as a share of that of a +//! sample of the front with as many points as the population; then the hypervolumes of the whole +//! front and of the sample. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example das_cmop2 +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{DasCmop2, MultiProblem}; +use genoxide::multi::{DifferentialEvolutionCrossover, MultiFitnessFunction}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points: 1.1 times the nadir point +const REFERENCE: [f64; 2] = [1.1, 1.1]; + +// the paper's population and evaluations: 300 for 1,000 generations +const POPULATION: usize = 300; +const GENERATIONS: u64 = 1_000; + +fn main() -> Result<()> { + let problem = DasCmop2::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the sample: as many points of the optimal front as the population has, about what a front + // of that many solutions can be + let sample = normalized(&problem, &problem.optimal_front(POPULATION).expect("known")); + let rate = 1.0 / problem.variables() as f64; + // MOEA/D-DE: a subproblem per weight vector, 30 neighbours (the paper's 0.1 N), parents from + // the neighbourhood with probability 0.2, differential evolution with F = 0.5 and CR = 1 + let moead = Moead::builder( + problem.representation(), + [Minimize; 2], + multi::das_dennis::<2>(POPULATION - 1), + ) + .neighbors(30) + .neighbor_mating(0.2) + .crossover(DifferentialEvolutionCrossover::new(0.5, 1.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run(&problem, "MOEA/D-DE", moead, &sample, &mut trace)?; + // the contrast: NSGA-II with the paper's settings + let nsga2 = Nsga2::builder(problem.representation(), [Minimize; 2]) + .population_size(POPULATION) + .crossover(SimulatedBinaryCrossover::new(20.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run(&problem, "NSGA-II", nsga2, &sample, &mut trace)?; + // the hypervolume of the whole front, from 20,000 of its points + let whole = normalized(&problem, &problem.optimal_front(20_000).expect("known")); + println!( + "the whole front: hypervolume {:.4}; a sample of {} of its points: {:.4}", + hypervolume(&whole, &REFERENCE, &[Minimize; 2]), + sample.len(), + hypervolume(&sample, &REFERENCE, &[Minimize; 2]) + ); + trace.write(&problem); + Ok(()) +} + +// runs `algorithm` for the paper's 1,000 generations, and prints its final front's size, how many +// of it are feasible, and its IGD+ to 500 points of the optimal front and hypervolume, with +// normalized objectives, as a share of the sample's +fn run( + problem: &DasCmop2, + name: &'static str, + algorithm: A, + sample: &[[f64; 2]], + trace: &mut trace::Trace, +) -> Result<()> +where + A: MultiObjectiveAlgorithm<2, Genome = Reals>, +{ + let mut record = trace.fronts(name); + let outcome = MultiEngine::new(algorithm, *problem) + .stop_when(Stop::generations(GENERATIONS)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 2], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 2]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "{name}, {GENERATIONS} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(500).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 2]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 2]); + let percent = 100.0 * volume / hypervolume(sample, &REFERENCE, &[Minimize; 2]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the \ + sample's", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &DasCmop2, points: &[[f64; 2]]) -> Vec<[f64; 2]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 2]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/das_cmop2/output.txt b/examples/das_cmop2/output.txt new file mode 100644 index 00000000..29fa1860 --- /dev/null +++ b/examples/das_cmop2/output.txt @@ -0,0 +1,3 @@ +MOEA/D-DE, 1000 generations: 266 solutions, 266 feasible, IGD+ 0.0017, hypervolume 0.8577, 99.8% of the sample's +NSGA-II, 1000 generations: 253 solutions, 253 feasible, IGD+ 0.1432, hypervolume 0.6155, 71.6% of the sample's +the whole front: hypervolume 0.8612; a sample of 300 of its points: 0.8597 diff --git a/examples/das_cmop2/trace.json b/examples/das_cmop2/trace.json new file mode 100644 index 00000000..5a6fbcbe --- /dev/null +++ b/examples/das_cmop2/trace.json @@ -0,0 +1,42 @@ 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+]} diff --git a/examples/das_cmop2/trace.py b/examples/das_cmop2/trace.py new file mode 100644 index 00000000..2aed8a3f --- /dev/null +++ b/examples/das_cmop2/trace.py @@ -0,0 +1,260 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: both fronts, each population's infeasible solutions (at most 100) and +feasible share, and the fronts' hypervolumes with normalized objectives, in at most 40 +generations; and the optimal front, in its pieces, and the feasible region, sampled. A frame's +evaluations are the two runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the samples of the feasible region along x₁ and along the distance variables, its grid, and the +# most infeasible solutions a frame shows +SAMPLES = 400 +DISTANCES = 800 +CELLS = 80 +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions, the hypervolume and the feasible share + self.series = {} + + def fronts(self, name): + """The callback for ``run`` that records the run ``name`` after each generation: None + without a trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + feasible = progress.front_violations == 0 + front = progress.front_objectives[feasible] + infeasible = progress.objectives[progress.violations != 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + share = (progress.violations == 0).sum() / len(progress.violations) + history.append( + (progress.evaluations, front.tolist(), infeasible.tolist(), volume, share) + ) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(40) + for generation in range(max(map(len, self.series.values()), default=0)): + at = {name: runs[min(generation, len(runs) - 1)] for name, runs in self.series.items()} + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at.values()), + "best": None, + "median": None, + "state": { + "fronts": {name: record[1] for name, record in at.items()}, + "infeasible": {name: record[2] for name, record in at.items()}, + "hypervolume": {name: record[3] for name, record in at.items()}, + "feasible": {name: record[4] for name, record in at.items()}, + }, + } + ) + whole = self.normalized(self.problem.optimal_front(20_000)) + settings = { + "format": 1, + "example": "das_cmop2", + "objective": ["minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "front-2d", + "problem": { + "objectives": ["f1", "f2"], + "true_front": pieces(self.problem.optimal_front(400).tolist()), + "series": list(self.series), + "feasible_region": region(self.problem), + }, + } + write(self.path, settings, frames.to_list()) + + +def pieces(front): + """The front in pieces, split where two points are more than 2% of its extent apart, each + point once.""" + first, last = front[0], front[-1] + extent = math.hypot(last[0] - first[0], first[1] - last[1]) + split = [] + for point in front: + if not split: + split.append([point]) + continue + previous = split[-1][-1] + if math.hypot(point[0] - previous[0], point[1] - previous[1]) > 0.02 * extent: + split.append([point]) + elif point != previous: + split[-1].append(point) + return split + + +def region(problem): + """The feasible region in objective space: genomes with x₁ evenly spread over [0, 1] and + every distance variable at a value v swept over [0, 1], which moves g from 0 up; a cell of a + grid of 80 × 80 over [0, 1.25 f₁ᵐᵃˣ] × [0, 1.25 f₂ᵐᵃˣ] (the nadir point's) is feasible if a + feasible genome lands in it, and the rows go up from f₂ = 0.""" + n = problem.dimensions + nadir = problem.nadir_point + width, height = 1.25 * nadir[0], 1.25 * nadir[1] + cells = [[False] * CELLS for _ in range(CELLS)] + for a in range(SAMPLES): + genomes = np.empty((DISTANCES, n)) + genomes[:, 0] = a / (SAMPLES - 1) + for b in range(DISTANCES): + genomes[b, 1:] = 0.0 + 1.0 * b / (DISTANCES - 1) + objectives, violations = problem.evaluate(genomes) + i = np.floor(objectives[:, 0] / width * CELLS) + j = np.floor(objectives[:, 1] / height * CELLS) + inside = (violations == 0) & (i >= 0) & (j >= 0) & (i < CELLS) & (j < CELLS) + for column, row in zip(i[inside].astype(int), j[inside].astype(int)): + cells[row][column] = True + # each row as the pairs of columns [start, end) where it's feasible + rows = [runs(row) for row in cells] + return {"x": [0.0, width], "y": [0.0, height], "columns": CELLS, "rows": rows} + + +def runs(row): + """The columns where a row of cells turns feasible or infeasible, as pairs [start, end).""" + turns = [ + column for column in range(len(row)) if row[column] != (column > 0 and row[column - 1]) + ] + return turns + ([len(row)] if row[-1] else []) + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/das_cmop2/trace.rs b/examples/das_cmop2/trace.rs new file mode 100644 index 00000000..1eca2583 --- /dev/null +++ b/examples/das_cmop2/trace.rs @@ -0,0 +1,367 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: both fronts, each population's infeasible solutions (at most 100) and +//! feasible share, and the fronts' hypervolumes with normalized objectives, in at most 40 +//! generations; and the optimal front, in its pieces, and the feasible region, sampled. A frame's +//! evaluations are the two runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{DasCmop2, MultiProblem}; +use genoxide::multi::{MultiFitnessFunction, MultiSnapshot, Scores}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, the +// hypervolume and the feasible share +type Record = (u64, Vec<[f64; 2]>, Vec<[f64; 2]>, f64, f64); + +// the samples of the feasible region along x₁ and along the distance variables, its grid, and the +// most infeasible solutions a frame shows +const SAMPLES: usize = 400; +const DISTANCES: usize = 800; +const CELLS: usize = 80; +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation + pub fn fronts(&mut self, name: &'static str) -> impl FnMut(&MultiSnapshot<'_, Reals, 2>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let values = |x: &Individual>| x.fitness()?.values(); + let feasible = |x: &&Individual>| { + x.fitness().is_some_and(|scores| scores.is_feasible()) + }; + let front: Vec<[f64; 2]> = snapshot + .front() + .iter() + .filter(feasible) + .filter_map(values) + .collect(); + let population = snapshot.population().as_slice(); + let infeasible = population.iter().filter(|x| !feasible(x)); + let infeasible: Vec<[f64; 2]> = infeasible.filter_map(values).take(SHOWN).collect(); + let share = population.iter().filter(feasible).count() as f64 / population.len() as f64; + let volume = hypervolume( + &normalized(&DasCmop2::default(), &front), + &REFERENCE, + &[Minimize; 2], + ); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume, share)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &DasCmop2) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(40); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let mut state = [Map::new(), Map::new(), Map::new(), Map::new()]; + let mut evaluations = 0; + for (name, history) in &self.series { + let (done, front, infeasible, volume, share) = + &history[generation.min(history.len() - 1)]; + let values = [json!(front), json!(infeasible), json!(volume), json!(share)]; + for (map, value) in state.iter_mut().zip(values) { + map.insert(name.to_string(), value); + } + evaluations += done; + } + let [fronts, infeasible, volumes, shares] = state; + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "state": { + "fronts": fronts, + "infeasible": infeasible, + "hypervolume": volumes, + "feasible": shares, + }, + })); + } + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(20_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "das_cmop2", + "objective": ["minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 2]), + "plot": "front-2d", + "problem": { + "objectives": ["f1", "f2"], + "true_front": pieces(&problem.optimal_front(400).expect("known")), + "series": names, + "feasible_region": region(problem), + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the front in pieces, split where two points are more than 2% of its extent apart, each point +// once +fn pieces(front: &[[f64; 2]]) -> Vec> { + let (first, last) = (front[0], front[front.len() - 1]); + let extent = (last[0] - first[0]).hypot(first[1] - last[1]); + let mut pieces: Vec> = Vec::new(); + for &point in front { + match pieces.last_mut() { + Some(piece) => { + let previous = piece[piece.len() - 1]; + if (point[0] - previous[0]).hypot(point[1] - previous[1]) > 0.02 * extent { + pieces.push(vec![point]); + } else if point != previous { + piece.push(point); + } + } + None => pieces.push(vec![point]), + } + } + pieces +} + +// the feasible region in objective space: genomes with x₁ evenly spread over [0, 1] and every +// distance variable at a value v swept over [0, 1], which moves g from 0 up; a cell of a grid +// of 80 × 80 over [0, 1.25 f₁ᵐᵃˣ] × [0, 1.25 f₂ᵐᵃˣ] (the nadir point's) is feasible if a feasible +// genome lands in it, and the rows go up from f₂ = 0 +fn region(problem: &DasCmop2) -> Value { + let n = problem.variables(); + let nadir = problem.nadir_point().expect("known"); + let (width, height) = (1.25 * nadir[0], 1.25 * nadir[1]); + let mut cells = vec![vec![false; CELLS]; CELLS]; + for a in 0..SAMPLES { + let mut x = vec![a as f64 / (SAMPLES - 1) as f64; 1]; + x.resize(n, 0.0); + for b in 0..DISTANCES { + let v = 0.0 + 1.0 * b as f64 / (DISTANCES - 1) as f64; + x[1..].fill(v); + let (f, violation) = problem.evaluate(&Reals::from(x.clone())); + let i = (f[0] / width * CELLS as f64).floor(); + let j = (f[1] / height * CELLS as f64).floor(); + if violation == 0.0 && i >= 0.0 && j >= 0.0 && i < CELLS as f64 && j < CELLS as f64 { + cells[j as usize][i as usize] = true; + } + } + } + // each row as the pairs of columns [start, end) where it's feasible + let rows: Vec> = cells.iter().map(Vec::as_slice).map(runs).collect(); + json!({"x": [0.0, width], "y": [0.0, height], "columns": CELLS, "rows": rows}) +} + +// the columns where a row of cells turns feasible or infeasible, as pairs [start, end) +fn runs(row: &[bool]) -> Vec { + let mut runs = Vec::new(); + for (column, &inside) in row.iter().enumerate() { + if inside != (column > 0 && row[column - 1]) { + runs.push(column); + } + } + if row.last() == Some(&true) { + runs.push(row.len()); + } + runs +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/das_cmop3/README.md b/examples/das_cmop3/README.md new file mode 100644 index 00000000..7f397099 --- /dev/null +++ b/examples/das_cmop3/README.md @@ -0,0 +1,74 @@ +--- +title: DAS-CMOP3 +category: multi-objective +summary: Minimize two objectives over 30 variables subject to 11 constraints of adjustable difficulty, whose Pareto front is disconnected pieces, with MOEA/D-DE, which reaches it in every run of 20, and NSGA-II, which doesn't, as the paper's doesn't. +reference: "Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). Difficulty adjustable and scalable constrained multiobjective test problem toolkit. Evolutionary Computation 28(3): 339-378." +reference_url: https://doi.org/10.1162/evco_a_00259 +optimum: "for the difficulty triplet (0, 0.5, 0.5), six pieces; ideal point (0.5, 0.5), nadir point (1.5, 1.5); hypervolume 0.7711 (normalized objectives, reference point (1.1, 1.1))" +languages: [rust, python] +order: 209 +family: DAS-CMOP +--- + +# DAS-CMOP3 + +## The problem + +Fan et al. (2020) built a toolkit of constrained test problems whose difficulty is set by three numbers, a triplet (η, ζ, γ) in [0, 1]³, one for each kind of difficulty: η for diversity, ζ for feasibility and γ for convergence. Each kind comes from a type of constraint. Type I constraints cut the front into pieces, narrower as η grows. A type II constraint asks the distance from the unconstrained front to be in a band, narrower as ζ grows. Type III constraints are infeasible regions in the objective space that block the way to the front, larger as γ grows. The paper suggests nine problems built this way, DAS-CMOP1 to DAS-CMOP9, and runs each with sixteen triplets (its table 3). + +DAS-CMOP3 (table 2) has two objectives over 30 variables in [0, 1]: + +```text +f₁ = x₁ + g +f₂ = 1 − √x₁ + 0.5 |sin(5πx₁)| + g +g = Σⱼ₌₂³⁰ (xⱼ − sin(0.5πx₁))² +subject to + sin(20πx₁) − b ≥ 0, b = 2η − 1 (type I) + (e − g)(g − 0.5) ≥ 0, e = 0.5 − ln ζ (type II) + ((f₁ − p_k) cos θ − (f₂ − q_k) sin θ)²/0.3 + + ((f₁ − p_k) sin θ + (f₂ − q_k) cos θ)²/1.2 ≥ r, r = γ/2 (type III, k = 1, …, 9) +θ = −π/4, (p_k, q_k) = (0, 1.5), (1, 0.5), (0, 2.5), (1, 1.5), (2, 0.5), (0, 3.5), (1, 2.5), (2, 1.5), (3, 0.5) +``` + +Both objectives are minimized. At g = 0 the unconstrained front is disconnected. The distance function g is 0 where every xⱼ equals sin(0.5πx₁): the variables are linked, each other variable's best value depending on x₁. The example uses the triplet with which the paper's figure 6 plots DAS-CMOP3, (0, 0.5, 0.5). Then the type I constraint has no effect (b = −1); g must be between 0.5 and 0.5 + ln 2 ≈ 1.193, which moves the front out along the diagonal by 0.5; and the nine ellipses, each 45° from the axes, have semi-axes √(0.3 r) ≈ 0.27 along the diagonal and √(1.2 r) ≈ 0.55 across it. + +The front depends on the triplet, and the paper samples it rather than writing it down; genoxide samples it in the same way, as the first feasible point of each of 20,000 rays α(x₁) + g (1, 1) that the type I constraint allows, non-dominated. With this triplet it is six pieces, the parts of the shifted curve's bumps that nothing dominates, f₁ from 0.5 to 0.504, 0.6498 to 0.7, 0.8782 to 0.9206, 1.0806 to 1.1284, 1.2822 to 1.3 and 1.4873 to 1.5, with f₁ from 0.897 to 1.128 on the boundary of the ellipse centered at (1, 0.5). Its ideal point is (0.5, 0.5) and its nadir point (1.5, 1.5). The authors published sampled fronts for the sixteen triplets with their code, which genoxide's agree with (the docs of `multi::problems::DasCmop1` say how they were compared). + +## What makes it hard + +The linked variables first. A solution on the front at x₁ has every other variable at sin(0.5πx₁): to move along the front, all 29 of them have to move together, by the same amount. Simulated binary crossover and polynomial mutation change each variable on its own, so a population that has settled on one value of x₁ stays there: any other x₁ costs 29 (Δ sin)² in g. Differential evolution moves them together: its steps are differences between solutions of the population, along the front once the population lies near it. The paper's MOEA/D-CDP uses it, and so does MOEA/D-DE here. Then the constraints: the band of g between 0.5 and e is feasible, a thin shell above the front, and the ellipses block parts of it. + +## Representation + +A `Real` genome of 30 genes in [0, 1]. The problem is genoxide's `DasCmop3`, whose fitness is the two objectives and the total constraint violation, 0 when it is feasible; the triplet is a `Difficulty`, `Difficulty::standard(k)` for the paper's sixteen. In Python, `gx.problems.DasCmop3()`, with `difficulty=(η, ζ, γ)` or the number of one of the sixteen, which `run` evaluates in Rust, so both versions print the same. + +Solutions compare by constrained dominance, the rule of the NSGA-II paper and the constraint handling that the paper calls CDP. A feasible solution beats an infeasible one. Of two infeasible ones, the smaller violation wins. Of two feasible ones, Pareto dominance decides. + +## Algorithm + +Two runs, each with the paper's population of 300 for 1,000 generations, 300,000 evaluations (section 7.1), and polynomial mutation with η = 20 at a rate of 1/30 per gene: + +- MOEA/D with differential evolution, MOEA/D-DE (Li and Zhang, 2009, IEEE Transactions on Evolutionary Computation 13(2): 284-302): a subproblem for each of 300 weight vectors spread evenly on the line w₁ + w₂ = 1, each the weighted Tchebycheff distance to the ideal point, with the paper's 30 neighbours (0.1 N) and at most 2 replacements per child; a child is its subproblem's solution moved by F = 0.5 times the difference of two parents, every gene (CR = 1), the parents from the neighbourhood with probability δ = 0.2; +- NSGA-II (Deb, Pratap, Agarwal and Meyarivan, 2002, IEEE Transactions on Evolutionary Computation 6(2): 182-197) with the paper's settings, simulated binary crossover with η = 20 at a rate of 0.9, as the contrast. + +Both compare solutions by constrained dominance. Differential evolution moves the linked variables together, which simulated binary crossover can't. + +δ = 0.2, where the paper and Li and Zhang use 0.9, keeps the population spread. While no solution is feasible, constrained dominance ranks solutions by their violation alone, and that pulls the whole population to one stretch of x₁: with seed 1 and δ = 0.9, 80% of the population has x₁ between 0.28 and 0.41 after 10 generations. Parents from the whole population, most of the time, spread it again once it is feasible; parents from the neighbourhood, nine times in ten, mostly can't, as the neighbourhood's solutions are all at that x₁. Over seeds 1 to 20, δ = 0.9 reaches the target in 1 runs of 20 (IGD+ from 0.0013 to 0.1614 and 67.8% to 99.8% of the sample's hypervolume). + +## Output + +A line per run: the size of its final front, how many of its solutions are feasible, the front's IGD+ and hypervolume, and the hypervolume as a share of that of a sample of 300 points of the optimal front, as many as the population. Then the hypervolumes of the whole optimal front and of the sample. + +Both indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 500 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest point of the found front, counting only the objectives in which the found point is worse. Smaller is better, and 0 means the found front covers the optimal one. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the area the front dominates up to the reference point (1.1, 1.1). Larger is better; the whole front's is computed from 20,000 of its points, and the sample's is what a front of 300 points reaches. Only feasible solutions count. + +The page plays both runs back over the grey feasible region, sampled from genomes with x₁ evenly spread and every distance variable at one value, swept; hollow points are infeasible solutions of the populations (at most 100 a frame), and the line is the optimal front, in its pieces. + +[The project page](https://tachsin.gr/projects/genoxide/examples/das-cmop3) plays this run back. + +## Good results + +The target: every solution feasible, and 99% of the hypervolume of the sample of 300 points of the front, about what a front of 300 solutions spread like it has. + +MOEA/D-DE's run with seed 1 ends with 147 solutions, 147 feasible, IGD+ 0.0010, hypervolume 0.7699, 99.9% of the sample's; NSGA-II's with 241 solutions, 241 feasible, IGD+ 0.1603, hypervolume 0.5241, 68.0% of the sample's. Over seeds 1 to 20, MOEA/D-DE ends with IGD+ from 0.0008 to 0.0011 and 99.8% to 99.9% of the sample's hypervolume, every run reaching the target, and NSGA-II with IGD+ from 0.1031 to 0.1831 and 65.8% to 79.0% of the sample's hypervolume, no run reaching it: NSGA-II converges to a short stretch of the front, or of the band above it, and stays there, for the reasons above. + +On DAS-CMOP3 the paper's NSGA-II-CDP ends with a mean IGD of 0.337 with the first triplet and 0.342 with (0.5, 0.5, 0.5) (table 4), where MOEA/D-CDP, with differential evolution, has 0.031 with the first. diff --git a/examples/das_cmop3/main.py b/examples/das_cmop3/main.py new file mode 100644 index 00000000..371cc915 --- /dev/null +++ b/examples/das_cmop3/main.py @@ -0,0 +1,101 @@ +"""DAS-CMOP3: minimize two objectives over 30 variables subject to 11 constraints, whose front is +disconnected, with MOEA/D-DE and NSGA-II. + +Fan et al.'s DAS-CMOP3, from genoxide's problems.DasCmop3, with the difficulty triplet of the paper's +figure 6, (0, 0.5, 0.5); run evaluates it in Rust. Runs MOEA/D with differential evolution +(MOEA/D-DE) and, as a contrast, NSGA-II with the paper's settings, a population of 300 for 1,000 +generations each (300,000 evaluations), with constraint dominance. Prints each final front's size, +how many of its solutions are feasible, and its IGD+ to 500 points of the optimal front and +hypervolume, with the objectives normalized by the front's ideal and nadir points, as a share of +that of a sample of the front with as many points as the population; then the hypervolumes of the +whole front and of the sample. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/das_cmop3/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points: 1.1 times the nadir point +REFERENCE = [1.1, 1.1] + +# the paper's population and evaluations: 300 for 1,000 generations +POPULATION = 300 +GENERATIONS = 1_000 + +problem = gx.problems.DasCmop3() +ideal, nadir = problem.ideal_point, problem.nadir_point +rate = 1 / problem.dimensions + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) +# the sample: as many points of the optimal front as the population has, about what a front of that +# many solutions can be +sample = normalized(problem.optimal_front(POPULATION)) + + +def run(name, algorithm): + """Runs ``algorithm`` for the paper's 1,000 generations, and prints its final front's size, how + many of it are feasible, and its IGD+ to 500 points of the optimal front and hypervolume, with + normalized objectives, as a share of the sample's.""" + result = algorithm.run(problem, generations=GENERATIONS, on_generation=trace.fronts(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"{name}, {GENERATIONS} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(500)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + percent = 100 * volume / gx.indicators.hypervolume(sample, REFERENCE) + print( + start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}, " + f"{percent:.1f}% of the sample's" + ) + + +# MOEA/D-DE: a subproblem per weight vector, 30 neighbours (the paper's 0.1 N), parents from the +# neighbourhood with probability 0.2, differential evolution with F = 0.5 and CR = 1 +moead = gx.Moead( + problem.genome, + objectives=problem.objectives, + weights=gx.das_dennis(2, POPULATION - 1), + neighbors=30, + neighbor_mating=0.2, + crossover=gx.DifferentialEvolutionCrossover(f=0.5, cr=1.0), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("MOEA/D-DE", moead) +# the contrast: NSGA-II with the paper's settings +nsga2 = gx.Nsga2( + problem.genome, + objectives=problem.objectives, + population_size=POPULATION, + crossover=gx.SimulatedBinaryCrossover(20), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-II", nsga2) +# the hypervolume of the whole front, from 20,000 of its points +whole = normalized(problem.optimal_front(20_000)) +print( + f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}; " + f"a sample of {len(sample)} of its points: " + f"{gx.indicators.hypervolume(sample, REFERENCE):.4f}" +) +trace.write() diff --git a/examples/das_cmop3/main.rs b/examples/das_cmop3/main.rs new file mode 100644 index 00000000..9fbd4ce1 --- /dev/null +++ b/examples/das_cmop3/main.rs @@ -0,0 +1,135 @@ +//! DAS-CMOP3: minimize two objectives over 30 variables subject to 11 constraints, whose front is +//! disconnected, with MOEA/D-DE and NSGA-II. +//! +//! Fan et al.'s DAS-CMOP3, from genoxide's `multi::problems::DasCmop3`, with the difficulty triplet of +//! the paper's figure 6, (0, 0.5, 0.5). Runs MOEA/D with differential evolution (MOEA/D-DE) and, +//! as a contrast, NSGA-II with the paper's settings, a population of 300 for 1,000 generations +//! each (300,000 evaluations), with constraint dominance. Prints each final front's size, how many +//! of its solutions are feasible, and its IGD+ to 500 points of the optimal front and hypervolume, +//! with the objectives normalized by the front's ideal and nadir points, as a share of that of a +//! sample of the front with as many points as the population; then the hypervolumes of the whole +//! front and of the sample. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example das_cmop3 +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{DasCmop3, MultiProblem}; +use genoxide::multi::{DifferentialEvolutionCrossover, MultiFitnessFunction}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points: 1.1 times the nadir point +const REFERENCE: [f64; 2] = [1.1, 1.1]; + +// the paper's population and evaluations: 300 for 1,000 generations +const POPULATION: usize = 300; +const GENERATIONS: u64 = 1_000; + +fn main() -> Result<()> { + let problem = DasCmop3::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the sample: as many points of the optimal front as the population has, about what a front + // of that many solutions can be + let sample = normalized(&problem, &problem.optimal_front(POPULATION).expect("known")); + let rate = 1.0 / problem.variables() as f64; + // MOEA/D-DE: a subproblem per weight vector, 30 neighbours (the paper's 0.1 N), parents from + // the neighbourhood with probability 0.2, differential evolution with F = 0.5 and CR = 1 + let moead = Moead::builder( + problem.representation(), + [Minimize; 2], + multi::das_dennis::<2>(POPULATION - 1), + ) + .neighbors(30) + .neighbor_mating(0.2) + .crossover(DifferentialEvolutionCrossover::new(0.5, 1.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run(&problem, "MOEA/D-DE", moead, &sample, &mut trace)?; + // the contrast: NSGA-II with the paper's settings + let nsga2 = Nsga2::builder(problem.representation(), [Minimize; 2]) + .population_size(POPULATION) + .crossover(SimulatedBinaryCrossover::new(20.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run(&problem, "NSGA-II", nsga2, &sample, &mut trace)?; + // the hypervolume of the whole front, from 20,000 of its points + let whole = normalized(&problem, &problem.optimal_front(20_000).expect("known")); + println!( + "the whole front: hypervolume {:.4}; a sample of {} of its points: {:.4}", + hypervolume(&whole, &REFERENCE, &[Minimize; 2]), + sample.len(), + hypervolume(&sample, &REFERENCE, &[Minimize; 2]) + ); + trace.write(&problem); + Ok(()) +} + +// runs `algorithm` for the paper's 1,000 generations, and prints its final front's size, how many +// of it are feasible, and its IGD+ to 500 points of the optimal front and hypervolume, with +// normalized objectives, as a share of the sample's +fn run( + problem: &DasCmop3, + name: &'static str, + algorithm: A, + sample: &[[f64; 2]], + trace: &mut trace::Trace, +) -> Result<()> +where + A: MultiObjectiveAlgorithm<2, Genome = Reals>, +{ + let mut record = trace.fronts(name); + let outcome = MultiEngine::new(algorithm, *problem) + .stop_when(Stop::generations(GENERATIONS)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 2], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 2]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "{name}, {GENERATIONS} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(500).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 2]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 2]); + let percent = 100.0 * volume / hypervolume(sample, &REFERENCE, &[Minimize; 2]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the \ + sample's", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &DasCmop3, points: &[[f64; 2]]) -> Vec<[f64; 2]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 2]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/das_cmop3/output.txt b/examples/das_cmop3/output.txt new file mode 100644 index 00000000..db91bee9 --- /dev/null +++ b/examples/das_cmop3/output.txt @@ -0,0 +1,3 @@ +MOEA/D-DE, 1000 generations: 147 solutions, 147 feasible, IGD+ 0.0010, hypervolume 0.7699, 99.9% of the sample's +NSGA-II, 1000 generations: 241 solutions, 241 feasible, IGD+ 0.1603, hypervolume 0.5241, 68.0% of the sample's +the whole front: hypervolume 0.7711; a sample of 300 of its points: 0.7709 diff --git a/examples/das_cmop3/trace.json b/examples/das_cmop3/trace.json new file mode 100644 index 00000000..3d7511a2 --- /dev/null +++ b/examples/das_cmop3/trace.json @@ -0,0 +1,31 @@ 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+]} diff --git a/examples/das_cmop3/trace.py b/examples/das_cmop3/trace.py new file mode 100644 index 00000000..67b23744 --- /dev/null +++ b/examples/das_cmop3/trace.py @@ -0,0 +1,260 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: both fronts, each population's infeasible solutions (at most 100) and +feasible share, and the fronts' hypervolumes with normalized objectives, in at most 40 +generations; and the optimal front, in its pieces, and the feasible region, sampled. A frame's +evaluations are the two runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the samples of the feasible region along x₁ and along the distance variables, its grid, and the +# most infeasible solutions a frame shows +SAMPLES = 400 +DISTANCES = 800 +CELLS = 80 +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions, the hypervolume and the feasible share + self.series = {} + + def fronts(self, name): + """The callback for ``run`` that records the run ``name`` after each generation: None + without a trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + feasible = progress.front_violations == 0 + front = progress.front_objectives[feasible] + infeasible = progress.objectives[progress.violations != 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + share = (progress.violations == 0).sum() / len(progress.violations) + history.append( + (progress.evaluations, front.tolist(), infeasible.tolist(), volume, share) + ) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(40) + for generation in range(max(map(len, self.series.values()), default=0)): + at = {name: runs[min(generation, len(runs) - 1)] for name, runs in self.series.items()} + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at.values()), + "best": None, + "median": None, + "state": { + "fronts": {name: record[1] for name, record in at.items()}, + "infeasible": {name: record[2] for name, record in at.items()}, + "hypervolume": {name: record[3] for name, record in at.items()}, + "feasible": {name: record[4] for name, record in at.items()}, + }, + } + ) + whole = self.normalized(self.problem.optimal_front(20_000)) + settings = { + "format": 1, + "example": "das_cmop3", + "objective": ["minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "front-2d", + "problem": { + "objectives": ["f1", "f2"], + "true_front": pieces(self.problem.optimal_front(400).tolist()), + "series": list(self.series), + "feasible_region": region(self.problem), + }, + } + write(self.path, settings, frames.to_list()) + + +def pieces(front): + """The front in pieces, split where two points are more than 2% of its extent apart, each + point once.""" + first, last = front[0], front[-1] + extent = math.hypot(last[0] - first[0], first[1] - last[1]) + split = [] + for point in front: + if not split: + split.append([point]) + continue + previous = split[-1][-1] + if math.hypot(point[0] - previous[0], point[1] - previous[1]) > 0.02 * extent: + split.append([point]) + elif point != previous: + split[-1].append(point) + return split + + +def region(problem): + """The feasible region in objective space: genomes with x₁ evenly spread over [0, 1] and + every distance variable at a value v swept over [0, 1], which moves g from 0 up; a cell of a + grid of 80 × 80 over [0, 1.25 f₁ᵐᵃˣ] × [0, 1.25 f₂ᵐᵃˣ] (the nadir point's) is feasible if a + feasible genome lands in it, and the rows go up from f₂ = 0.""" + n = problem.dimensions + nadir = problem.nadir_point + width, height = 1.25 * nadir[0], 1.25 * nadir[1] + cells = [[False] * CELLS for _ in range(CELLS)] + for a in range(SAMPLES): + genomes = np.empty((DISTANCES, n)) + genomes[:, 0] = a / (SAMPLES - 1) + for b in range(DISTANCES): + genomes[b, 1:] = 0.0 + 1.0 * b / (DISTANCES - 1) + objectives, violations = problem.evaluate(genomes) + i = np.floor(objectives[:, 0] / width * CELLS) + j = np.floor(objectives[:, 1] / height * CELLS) + inside = (violations == 0) & (i >= 0) & (j >= 0) & (i < CELLS) & (j < CELLS) + for column, row in zip(i[inside].astype(int), j[inside].astype(int)): + cells[row][column] = True + # each row as the pairs of columns [start, end) where it's feasible + rows = [runs(row) for row in cells] + return {"x": [0.0, width], "y": [0.0, height], "columns": CELLS, "rows": rows} + + +def runs(row): + """The columns where a row of cells turns feasible or infeasible, as pairs [start, end).""" + turns = [ + column for column in range(len(row)) if row[column] != (column > 0 and row[column - 1]) + ] + return turns + ([len(row)] if row[-1] else []) + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/das_cmop3/trace.rs b/examples/das_cmop3/trace.rs new file mode 100644 index 00000000..ac7b68b1 --- /dev/null +++ b/examples/das_cmop3/trace.rs @@ -0,0 +1,367 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: both fronts, each population's infeasible solutions (at most 100) and +//! feasible share, and the fronts' hypervolumes with normalized objectives, in at most 40 +//! generations; and the optimal front, in its pieces, and the feasible region, sampled. A frame's +//! evaluations are the two runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{DasCmop3, MultiProblem}; +use genoxide::multi::{MultiFitnessFunction, MultiSnapshot, Scores}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, the +// hypervolume and the feasible share +type Record = (u64, Vec<[f64; 2]>, Vec<[f64; 2]>, f64, f64); + +// the samples of the feasible region along x₁ and along the distance variables, its grid, and the +// most infeasible solutions a frame shows +const SAMPLES: usize = 400; +const DISTANCES: usize = 800; +const CELLS: usize = 80; +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation + pub fn fronts(&mut self, name: &'static str) -> impl FnMut(&MultiSnapshot<'_, Reals, 2>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let values = |x: &Individual>| x.fitness()?.values(); + let feasible = |x: &&Individual>| { + x.fitness().is_some_and(|scores| scores.is_feasible()) + }; + let front: Vec<[f64; 2]> = snapshot + .front() + .iter() + .filter(feasible) + .filter_map(values) + .collect(); + let population = snapshot.population().as_slice(); + let infeasible = population.iter().filter(|x| !feasible(x)); + let infeasible: Vec<[f64; 2]> = infeasible.filter_map(values).take(SHOWN).collect(); + let share = population.iter().filter(feasible).count() as f64 / population.len() as f64; + let volume = hypervolume( + &normalized(&DasCmop3::default(), &front), + &REFERENCE, + &[Minimize; 2], + ); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume, share)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &DasCmop3) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(40); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let mut state = [Map::new(), Map::new(), Map::new(), Map::new()]; + let mut evaluations = 0; + for (name, history) in &self.series { + let (done, front, infeasible, volume, share) = + &history[generation.min(history.len() - 1)]; + let values = [json!(front), json!(infeasible), json!(volume), json!(share)]; + for (map, value) in state.iter_mut().zip(values) { + map.insert(name.to_string(), value); + } + evaluations += done; + } + let [fronts, infeasible, volumes, shares] = state; + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "state": { + "fronts": fronts, + "infeasible": infeasible, + "hypervolume": volumes, + "feasible": shares, + }, + })); + } + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(20_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "das_cmop3", + "objective": ["minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 2]), + "plot": "front-2d", + "problem": { + "objectives": ["f1", "f2"], + "true_front": pieces(&problem.optimal_front(400).expect("known")), + "series": names, + "feasible_region": region(problem), + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the front in pieces, split where two points are more than 2% of its extent apart, each point +// once +fn pieces(front: &[[f64; 2]]) -> Vec> { + let (first, last) = (front[0], front[front.len() - 1]); + let extent = (last[0] - first[0]).hypot(first[1] - last[1]); + let mut pieces: Vec> = Vec::new(); + for &point in front { + match pieces.last_mut() { + Some(piece) => { + let previous = piece[piece.len() - 1]; + if (point[0] - previous[0]).hypot(point[1] - previous[1]) > 0.02 * extent { + pieces.push(vec![point]); + } else if point != previous { + piece.push(point); + } + } + None => pieces.push(vec![point]), + } + } + pieces +} + +// the feasible region in objective space: genomes with x₁ evenly spread over [0, 1] and every +// distance variable at a value v swept over [0, 1], which moves g from 0 up; a cell of a grid +// of 80 × 80 over [0, 1.25 f₁ᵐᵃˣ] × [0, 1.25 f₂ᵐᵃˣ] (the nadir point's) is feasible if a feasible +// genome lands in it, and the rows go up from f₂ = 0 +fn region(problem: &DasCmop3) -> Value { + let n = problem.variables(); + let nadir = problem.nadir_point().expect("known"); + let (width, height) = (1.25 * nadir[0], 1.25 * nadir[1]); + let mut cells = vec![vec![false; CELLS]; CELLS]; + for a in 0..SAMPLES { + let mut x = vec![a as f64 / (SAMPLES - 1) as f64; 1]; + x.resize(n, 0.0); + for b in 0..DISTANCES { + let v = 0.0 + 1.0 * b as f64 / (DISTANCES - 1) as f64; + x[1..].fill(v); + let (f, violation) = problem.evaluate(&Reals::from(x.clone())); + let i = (f[0] / width * CELLS as f64).floor(); + let j = (f[1] / height * CELLS as f64).floor(); + if violation == 0.0 && i >= 0.0 && j >= 0.0 && i < CELLS as f64 && j < CELLS as f64 { + cells[j as usize][i as usize] = true; + } + } + } + // each row as the pairs of columns [start, end) where it's feasible + let rows: Vec> = cells.iter().map(Vec::as_slice).map(runs).collect(); + json!({"x": [0.0, width], "y": [0.0, height], "columns": CELLS, "rows": rows}) +} + +// the columns where a row of cells turns feasible or infeasible, as pairs [start, end) +fn runs(row: &[bool]) -> Vec { + let mut runs = Vec::new(); + for (column, &inside) in row.iter().enumerate() { + if inside != (column > 0 && row[column - 1]) { + runs.push(column); + } + } + if row.last() == Some(&true) { + runs.push(row.len()); + } + runs +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/das_cmop4/README.md b/examples/das_cmop4/README.md new file mode 100644 index 00000000..f84ea27a --- /dev/null +++ b/examples/das_cmop4/README.md @@ -0,0 +1,72 @@ +--- +title: DAS-CMOP4 +category: multi-objective +summary: Minimize two objectives over 30 variables subject to 11 constraints of adjustable difficulty, whose Pareto front is pieces of a concave curve, with NSGA-II. +reference: "Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). Difficulty adjustable and scalable constrained multiobjective test problem toolkit. Evolutionary Computation 28(3): 339-378." +reference_url: https://doi.org/10.1162/evco_a_00259 +optimum: "for the difficulty triplet (0.5, 0.5, 0.5), six pieces of the curve f₂ = 1.5 − (f₁ − 0.5)²; ideal point (0.5, 0.5975), nadir point (1.45, 1.5); hypervolume 0.4476 (normalized objectives, reference point (1.1, 1.1))" +languages: [rust, python] +order: 210 +family: DAS-CMOP +--- + +# DAS-CMOP4 + +## The problem + +Fan et al. (2020) built a toolkit of constrained test problems whose difficulty is set by three numbers, a triplet (η, ζ, γ) in [0, 1]³, one for each kind of difficulty: η for diversity, ζ for feasibility and γ for convergence. Each kind comes from a type of constraint. Type I constraints cut the front into pieces, narrower as η grows. A type II constraint asks the distance from the unconstrained front to be in a band, narrower as ζ grows. Type III constraints are infeasible regions in the objective space that block the way to the front, larger as γ grows. The paper suggests nine problems built this way, DAS-CMOP1 to DAS-CMOP9, and runs each with sixteen triplets (its table 3). + +DAS-CMOP4 (table 2) has two objectives over 30 variables in [0, 1]: + +```text +f₁ = x₁ + g +f₂ = 1 − x₁² + g +g = 29 + Σⱼ₌₂³⁰ ((xⱼ − 0.5)² − cos(20π(xⱼ − 0.5))) +subject to + sin(20πx₁) − b ≥ 0, b = 2η − 1 (type I) + (e − g)(g − 0.5) ≥ 0, e = 0.5 − ln ζ (type II) + ((f₁ − p_k) cos θ − (f₂ − q_k) sin θ)²/0.3 + + ((f₁ − p_k) sin θ + (f₂ − q_k) cos θ)²/1.2 ≥ r, r = γ/2 (type III, k = 1, …, 9) +θ = −π/4, (p_k, q_k) = (0, 1.5), (1, 0.5), (0, 2.5), (1, 1.5), (2, 0.5), (0, 3.5), (1, 2.5), (2, 1.5), (3, 0.5) +``` + +Both objectives are minimized. At g = 0 the unconstrained front is concave. The distance function g is 0 with every xⱼ at 0.5, and has 11²⁹ − 1 local minima, the cosine's, as DTLZ1's. The example uses the triplet with which the paper's figure 6 plots DAS-CMOP4, (0.5, 0.5, 0.5). Then b = 0: sin(20πx₁) ≥ 0 keeps x₁ in [0, 0.05], [0.1, 0.15], …, half of it; g must be between 0.5 and 0.5 + ln 2 ≈ 1.193, which moves the front out along the diagonal by 0.5; and the nine ellipses, each 45° from the axes, have semi-axes √(0.3 r) ≈ 0.27 along the diagonal and √(1.2 r) ≈ 0.55 across it. + +The front depends on the triplet, and the paper samples it rather than writing it down; genoxide samples it in the same way, as the first feasible point of each of 20,000 rays α(x₁) + g (1, 1) that the type I constraint allows, non-dominated. With this triplet it is six pieces of the shifted concave curve, f₁ from 0.5 to 0.55, 0.6 to 0.65, 1.1424 to 1.15, 1.2 to 1.25, 1.3 to 1.35 and 1.4 to 1.45: the type I constraint cuts x₁ into ten intervals of width 0.05, and the ellipse centered at (1, 1.5) takes out the middle ones. Its ideal point is (0.5, 0.5975) and its nadir point (1.45, 1.5). The authors published sampled fronts for the sixteen triplets with their code, which genoxide's agree with (the docs of `multi::problems::DasCmop1` say how they were compared). + +## What makes it hard + +The multimodal g first: its local minima put local fronts parallel to the true one, and the type II band, g between 0.5 and e, is itself a target among them: a population that reaches the band at a local minimum of g must leave the band to improve, through infeasible solutions. Then the type I constraint, which cuts the front into short pieces that a population must spread over, and the ellipses, which block the way to some of them: a population that converges behind one stays there, on the far side, with constraint dominance, which never accepts an infeasible step. + +## Representation + +A `Real` genome of 30 genes in [0, 1]. The problem is genoxide's `DasCmop4`, whose fitness is the two objectives and the total constraint violation, 0 when it is feasible; the triplet is a `Difficulty`, `Difficulty::standard(k)` for the paper's sixteen. In Python, `gx.problems.DasCmop4()`, with `difficulty=(η, ζ, γ)` or the number of one of the sixteen, which `run` evaluates in Rust, so both versions print the same. + +Solutions compare by constrained dominance, the rule of the NSGA-II paper and the constraint handling that the paper calls CDP. A feasible solution beats an infeasible one. Of two infeasible ones, the smaller violation wins. Of two feasible ones, Pareto dominance decides. + +## Algorithm + +NSGA-II (Deb, Pratap, Agarwal and Meyarivan, 2002, IEEE Transactions on Evolutionary Computation 6(2): 182-197) with the paper's settings: a population of 300 for 1,000 generations, 300,000 evaluations (section 7.1), simulated binary crossover with η = 20 at a rate of 0.9, and polynomial mutation at a rate of 1/30 per gene, twice: + +- with η = 20 for the mutation, the paper's; +- with η = 5, whose steps are larger: the median step is about 3% of the range with η = 20 and about 11% with η = 5. + +The larger steps of η = 5 keep the population spread over x₁ while it converges, so that it reaches every piece. + +## Output + +A line per run: the size of its final front, how many of its solutions are feasible, and the front's IGD+ and hypervolume. Then the hypervolume of the whole optimal front. + +Both indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 500 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest point of the found front, counting only the objectives in which the found point is worse. Smaller is better, and 0 means the found front covers the optimal one. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the area the front dominates up to the reference point (1.1, 1.1). Larger is better; the whole front's is computed from 20,000 of its points. Only feasible solutions count. + +The page plays both runs back over the grey feasible region, sampled from genomes with x₁ evenly spread and every distance variable at one value, swept; hollow points are infeasible solutions of the populations (at most 100 a frame), and the line is the optimal front, in its pieces. + +[The project page](https://tachsin.gr/projects/genoxide/examples/das-cmop4) plays this run back. + +## Good results + +The target: an IGD+ of at most 0.01 in normalized objectives, with every solution feasible. + +With the paper's settings, the run with seed 1 ends with 300 solutions, 300 feasible, IGD+ 0.0155, hypervolume 0.4201. With η = 5, 300 solutions, 300 feasible, IGD+ 0.0005, hypervolume 0.4459. Over seeds 1 to 20, with η = 20, 12 runs reach the target; the median IGD+ is 0.0006, the worst 0.2721; with η = 5, every run reaches the target, with IGD+ from 0.0002 to 0.0014. + +The paper's NSGA-II-CDP ends with a mean IGD of 0.111 on DAS-CMOP4 with this triplet (its table 4, number 8), with a standard deviation of 0.157: some runs reach the front and some don't, as here with η = 20; MOEA/D-CDP has 0.204. diff --git a/examples/das_cmop4/main.py b/examples/das_cmop4/main.py new file mode 100644 index 00000000..9593802c --- /dev/null +++ b/examples/das_cmop4/main.py @@ -0,0 +1,77 @@ +"""DAS-CMOP4: minimize two objectives over 30 variables subject to 11 constraints, whose front is pieces of a concave curve, +with NSGA-II. + +Fan et al.'s DAS-CMOP4, from genoxide's problems.DasCmop4, with the difficulty triplet of the paper's +figure 6, (0.5, 0.5, 0.5); run evaluates it in Rust. Runs NSGA-II twice with the paper's settings, a +population of 300 for 1,000 generations (300,000 evaluations): with polynomial mutation with +η = 20, as the paper's, and with η = 5. Prints each final front's size, how many of its solutions +are feasible, and its IGD+ to 500 points of the optimal front and hypervolume, with the objectives +normalized by the front's ideal and nadir points; then the whole front's hypervolume. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/das_cmop4/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points: 1.1 times the nadir point +REFERENCE = [1.1, 1.1] + +# the paper's population and evaluations: 300 for 1,000 generations +POPULATION = 300 +GENERATIONS = 1_000 + +problem = gx.problems.DasCmop4() +ideal, nadir = problem.ideal_point, problem.nadir_point + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) + + +def run(name, eta): + """Runs NSGA-II with the paper's population, simulated binary crossover with η = 20 at + genoxide's default rate of 0.9, and polynomial mutation with the distribution index ``eta`` + at a rate of 1/n per gene; prints its final front's size, how many of it are feasible, and its + IGD+ to 500 points of the optimal front and hypervolume, with normalized objectives.""" + nsga2 = gx.Nsga2( + problem.genome, + objectives=problem.objectives, + population_size=POPULATION, + crossover=gx.SimulatedBinaryCrossover(20), + mutation=gx.PolynomialMutation(eta, rate=1 / problem.dimensions), + seed=1, + ) + result = nsga2.run(problem, generations=GENERATIONS, on_generation=trace.fronts(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"NSGA-II, {name}, {GENERATIONS} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(500)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + print(start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}") + + +# the paper's settings: polynomial mutation with η = 20 +run("η = 20", 20) +# polynomial mutation with η = 5, whose steps are larger +run("η = 5", 5) +# the hypervolume of the whole front, from 20,000 of its points +whole = normalized(problem.optimal_front(20_000)) +print(f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}") +trace.write() diff --git a/examples/das_cmop4/main.rs b/examples/das_cmop4/main.rs new file mode 100644 index 00000000..2ed9ce08 --- /dev/null +++ b/examples/das_cmop4/main.rs @@ -0,0 +1,104 @@ +//! DAS-CMOP4: minimize two objectives over 30 variables subject to 11 constraints, whose front is pieces of a concave curve, with +//! NSGA-II. +//! +//! Fan et al.'s DAS-CMOP4, from genoxide's `multi::problems::DasCmop4`, with the difficulty triplet of +//! the paper's figure 6, (0.5, 0.5, 0.5). Runs NSGA-II twice with the paper's settings, a population +//! of 300 for 1,000 generations (300,000 evaluations): with polynomial mutation with η = 20, as +//! the paper's, and with η = 5. Prints each final front's size, how many of its solutions are +//! feasible, and its IGD+ to 500 points of the optimal front and hypervolume, with the objectives +//! normalized by the front's ideal and nadir points; then the whole front's hypervolume. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example das_cmop4 +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{DasCmop4, MultiProblem}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points: 1.1 times the nadir point +const REFERENCE: [f64; 2] = [1.1, 1.1]; + +// the paper's population and evaluations: 300 for 1,000 generations +const POPULATION: usize = 300; +const GENERATIONS: u64 = 1_000; + +fn main() -> Result<()> { + let problem = DasCmop4::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the paper's settings: polynomial mutation with η = 20 + run(&problem, "η = 20", 20.0, &mut trace)?; + // polynomial mutation with η = 5, whose steps are larger + run(&problem, "η = 5", 5.0, &mut trace)?; + // the hypervolume of the whole front, from 20,000 of its points + let whole = normalized(&problem, &problem.optimal_front(20_000).expect("known")); + let volume = hypervolume(&whole, &REFERENCE, &[Minimize; 2]); + println!("the whole front: hypervolume {volume:.4}"); + trace.write(&problem); + Ok(()) +} + +// runs NSGA-II with the paper's population, simulated binary crossover with η = 20 at genoxide's +// default rate of 0.9, and polynomial mutation with the distribution index `eta` at a rate of 1/n +// per gene; prints its final front's size, how many of it are feasible, and its IGD+ to 500 +// points of the optimal front and hypervolume, with normalized objectives +fn run(problem: &DasCmop4, name: &'static str, eta: f64, trace: &mut trace::Trace) -> Result<()> { + let rate = 1.0 / problem.variables() as f64; + let nsga2 = Nsga2::builder(problem.representation(), [Minimize; 2]) + .population_size(POPULATION) + .crossover(SimulatedBinaryCrossover::new(20.0)?) + .mutate(PolynomialMutation::per_gene(rate, eta)?) + .seed(1) + .build()?; + let mut record = trace.fronts(name); + let outcome = MultiEngine::new(nsga2, *problem) + .stop_when(Stop::generations(GENERATIONS)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 2], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 2]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "NSGA-II, {name}, {GENERATIONS} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(500).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 2]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 2]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &DasCmop4, points: &[[f64; 2]]) -> Vec<[f64; 2]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 2]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/das_cmop4/output.txt b/examples/das_cmop4/output.txt new file mode 100644 index 00000000..4d6fdd09 --- /dev/null +++ b/examples/das_cmop4/output.txt @@ -0,0 +1,3 @@ +NSGA-II, η = 20, 1000 generations: 300 solutions, 300 feasible, IGD+ 0.0155, hypervolume 0.4201 +NSGA-II, η = 5, 1000 generations: 300 solutions, 300 feasible, IGD+ 0.0005, hypervolume 0.4459 +the whole front: hypervolume 0.4476 diff --git a/examples/das_cmop4/trace.json b/examples/das_cmop4/trace.json new file mode 100644 index 00000000..1feee6a6 --- /dev/null +++ b/examples/das_cmop4/trace.json @@ -0,0 +1,42 @@ 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= 20":0.415951,"η = 5":0.445647},"infeasible":{"η = 20":[],"η = 5":[]}}} +]} diff --git a/examples/das_cmop4/trace.py b/examples/das_cmop4/trace.py new file mode 100644 index 00000000..45428a9f --- /dev/null +++ b/examples/das_cmop4/trace.py @@ -0,0 +1,260 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: both fronts, each population's infeasible solutions (at most 100) and +feasible share, and the fronts' hypervolumes with normalized objectives, in at most 40 +generations; and the optimal front, in its pieces, and the feasible region, sampled. A frame's +evaluations are the two runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the samples of the feasible region along x₁ and along the distance variables, its grid, and the +# most infeasible solutions a frame shows +SAMPLES = 400 +DISTANCES = 800 +CELLS = 80 +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions, the hypervolume and the feasible share + self.series = {} + + def fronts(self, name): + """The callback for ``run`` that records the run ``name`` after each generation: None + without a trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + feasible = progress.front_violations == 0 + front = progress.front_objectives[feasible] + infeasible = progress.objectives[progress.violations != 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + share = (progress.violations == 0).sum() / len(progress.violations) + history.append( + (progress.evaluations, front.tolist(), infeasible.tolist(), volume, share) + ) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(40) + for generation in range(max(map(len, self.series.values()), default=0)): + at = {name: runs[min(generation, len(runs) - 1)] for name, runs in self.series.items()} + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at.values()), + "best": None, + "median": None, + "state": { + "fronts": {name: record[1] for name, record in at.items()}, + "infeasible": {name: record[2] for name, record in at.items()}, + "hypervolume": {name: record[3] for name, record in at.items()}, + "feasible": {name: record[4] for name, record in at.items()}, + }, + } + ) + whole = self.normalized(self.problem.optimal_front(20_000)) + settings = { + "format": 1, + "example": "das_cmop4", + "objective": ["minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "front-2d", + "problem": { + "objectives": ["f1", "f2"], + "true_front": pieces(self.problem.optimal_front(400).tolist()), + "series": list(self.series), + "feasible_region": region(self.problem), + }, + } + write(self.path, settings, frames.to_list()) + + +def pieces(front): + """The front in pieces, split where two points are more than 2% of its extent apart, each + point once.""" + first, last = front[0], front[-1] + extent = math.hypot(last[0] - first[0], first[1] - last[1]) + split = [] + for point in front: + if not split: + split.append([point]) + continue + previous = split[-1][-1] + if math.hypot(point[0] - previous[0], point[1] - previous[1]) > 0.02 * extent: + split.append([point]) + elif point != previous: + split[-1].append(point) + return split + + +def region(problem): + """The feasible region in objective space: genomes with x₁ evenly spread over [0, 1] and + every distance variable at a value v swept over [0.5, 0.52], which moves g from 0 up; a cell of a + grid of 80 × 80 over [0, 1.25 f₁ᵐᵃˣ] × [0, 1.25 f₂ᵐᵃˣ] (the nadir point's) is feasible if a + feasible genome lands in it, and the rows go up from f₂ = 0.""" + n = problem.dimensions + nadir = problem.nadir_point + width, height = 1.25 * nadir[0], 1.25 * nadir[1] + cells = [[False] * CELLS for _ in range(CELLS)] + for a in range(SAMPLES): + genomes = np.empty((DISTANCES, n)) + genomes[:, 0] = a / (SAMPLES - 1) + for b in range(DISTANCES): + genomes[b, 1:] = 0.5 + 0.02 * b / (DISTANCES - 1) + objectives, violations = problem.evaluate(genomes) + i = np.floor(objectives[:, 0] / width * CELLS) + j = np.floor(objectives[:, 1] / height * CELLS) + inside = (violations == 0) & (i >= 0) & (j >= 0) & (i < CELLS) & (j < CELLS) + for column, row in zip(i[inside].astype(int), j[inside].astype(int)): + cells[row][column] = True + # each row as the pairs of columns [start, end) where it's feasible + rows = [runs(row) for row in cells] + return {"x": [0.0, width], "y": [0.0, height], "columns": CELLS, "rows": rows} + + +def runs(row): + """The columns where a row of cells turns feasible or infeasible, as pairs [start, end).""" + turns = [ + column for column in range(len(row)) if row[column] != (column > 0 and row[column - 1]) + ] + return turns + ([len(row)] if row[-1] else []) + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/das_cmop4/trace.rs b/examples/das_cmop4/trace.rs new file mode 100644 index 00000000..d55ea059 --- /dev/null +++ b/examples/das_cmop4/trace.rs @@ -0,0 +1,367 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: both fronts, each population's infeasible solutions (at most 100) and +//! feasible share, and the fronts' hypervolumes with normalized objectives, in at most 40 +//! generations; and the optimal front, in its pieces, and the feasible region, sampled. A frame's +//! evaluations are the two runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{DasCmop4, MultiProblem}; +use genoxide::multi::{MultiFitnessFunction, MultiSnapshot, Scores}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, the +// hypervolume and the feasible share +type Record = (u64, Vec<[f64; 2]>, Vec<[f64; 2]>, f64, f64); + +// the samples of the feasible region along x₁ and along the distance variables, its grid, and the +// most infeasible solutions a frame shows +const SAMPLES: usize = 400; +const DISTANCES: usize = 800; +const CELLS: usize = 80; +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation + pub fn fronts(&mut self, name: &'static str) -> impl FnMut(&MultiSnapshot<'_, Reals, 2>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let values = |x: &Individual>| x.fitness()?.values(); + let feasible = |x: &&Individual>| { + x.fitness().is_some_and(|scores| scores.is_feasible()) + }; + let front: Vec<[f64; 2]> = snapshot + .front() + .iter() + .filter(feasible) + .filter_map(values) + .collect(); + let population = snapshot.population().as_slice(); + let infeasible = population.iter().filter(|x| !feasible(x)); + let infeasible: Vec<[f64; 2]> = infeasible.filter_map(values).take(SHOWN).collect(); + let share = population.iter().filter(feasible).count() as f64 / population.len() as f64; + let volume = hypervolume( + &normalized(&DasCmop4::default(), &front), + &REFERENCE, + &[Minimize; 2], + ); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume, share)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &DasCmop4) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(40); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let mut state = [Map::new(), Map::new(), Map::new(), Map::new()]; + let mut evaluations = 0; + for (name, history) in &self.series { + let (done, front, infeasible, volume, share) = + &history[generation.min(history.len() - 1)]; + let values = [json!(front), json!(infeasible), json!(volume), json!(share)]; + for (map, value) in state.iter_mut().zip(values) { + map.insert(name.to_string(), value); + } + evaluations += done; + } + let [fronts, infeasible, volumes, shares] = state; + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "state": { + "fronts": fronts, + "infeasible": infeasible, + "hypervolume": volumes, + "feasible": shares, + }, + })); + } + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(20_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "das_cmop4", + "objective": ["minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 2]), + "plot": "front-2d", + "problem": { + "objectives": ["f1", "f2"], + "true_front": pieces(&problem.optimal_front(400).expect("known")), + "series": names, + "feasible_region": region(problem), + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the front in pieces, split where two points are more than 2% of its extent apart, each point +// once +fn pieces(front: &[[f64; 2]]) -> Vec> { + let (first, last) = (front[0], front[front.len() - 1]); + let extent = (last[0] - first[0]).hypot(first[1] - last[1]); + let mut pieces: Vec> = Vec::new(); + for &point in front { + match pieces.last_mut() { + Some(piece) => { + let previous = piece[piece.len() - 1]; + if (point[0] - previous[0]).hypot(point[1] - previous[1]) > 0.02 * extent { + pieces.push(vec![point]); + } else if point != previous { + piece.push(point); + } + } + None => pieces.push(vec![point]), + } + } + pieces +} + +// the feasible region in objective space: genomes with x₁ evenly spread over [0, 1] and every +// distance variable at a value v swept over [0.5, 0.52], which moves g from 0 up; a cell of a grid +// of 80 × 80 over [0, 1.25 f₁ᵐᵃˣ] × [0, 1.25 f₂ᵐᵃˣ] (the nadir point's) is feasible if a feasible +// genome lands in it, and the rows go up from f₂ = 0 +fn region(problem: &DasCmop4) -> Value { + let n = problem.variables(); + let nadir = problem.nadir_point().expect("known"); + let (width, height) = (1.25 * nadir[0], 1.25 * nadir[1]); + let mut cells = vec![vec![false; CELLS]; CELLS]; + for a in 0..SAMPLES { + let mut x = vec![a as f64 / (SAMPLES - 1) as f64; 1]; + x.resize(n, 0.0); + for b in 0..DISTANCES { + let v = 0.5 + 0.02 * b as f64 / (DISTANCES - 1) as f64; + x[1..].fill(v); + let (f, violation) = problem.evaluate(&Reals::from(x.clone())); + let i = (f[0] / width * CELLS as f64).floor(); + let j = (f[1] / height * CELLS as f64).floor(); + if violation == 0.0 && i >= 0.0 && j >= 0.0 && i < CELLS as f64 && j < CELLS as f64 { + cells[j as usize][i as usize] = true; + } + } + } + // each row as the pairs of columns [start, end) where it's feasible + let rows: Vec> = cells.iter().map(Vec::as_slice).map(runs).collect(); + json!({"x": [0.0, width], "y": [0.0, height], "columns": CELLS, "rows": rows}) +} + +// the columns where a row of cells turns feasible or infeasible, as pairs [start, end) +fn runs(row: &[bool]) -> Vec { + let mut runs = Vec::new(); + for (column, &inside) in row.iter().enumerate() { + if inside != (column > 0 && row[column - 1]) { + runs.push(column); + } + } + if row.last() == Some(&true) { + runs.push(row.len()); + } + runs +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/das_cmop5/README.md b/examples/das_cmop5/README.md new file mode 100644 index 00000000..a29129d3 --- /dev/null +++ b/examples/das_cmop5/README.md @@ -0,0 +1,72 @@ +--- +title: DAS-CMOP5 +category: multi-objective +summary: Minimize two objectives over 30 variables subject to 11 constraints of adjustable difficulty, whose Pareto front is ten pieces of a convex curve, with NSGA-II. +reference: "Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). Difficulty adjustable and scalable constrained multiobjective test problem toolkit. Evolutionary Computation 28(3): 339-378." +reference_url: https://doi.org/10.1162/evco_a_00259 +optimum: "for the difficulty triplet (0.5, 0.5, 0.5), ten pieces, of the curve f₂ = 1.5 − √(f₁ − 0.5) and of an ellipse; ideal point (0.5, 0.5253), nadir point (1.45, 1.5); hypervolume 0.8494 (normalized objectives, reference point (1.1, 1.1))" +languages: [rust, python] +order: 211 +family: DAS-CMOP +--- + +# DAS-CMOP5 + +## The problem + +Fan et al. (2020) built a toolkit of constrained test problems whose difficulty is set by three numbers, a triplet (η, ζ, γ) in [0, 1]³, one for each kind of difficulty: η for diversity, ζ for feasibility and γ for convergence. Each kind comes from a type of constraint. Type I constraints cut the front into pieces, narrower as η grows. A type II constraint asks the distance from the unconstrained front to be in a band, narrower as ζ grows. Type III constraints are infeasible regions in the objective space that block the way to the front, larger as γ grows. The paper suggests nine problems built this way, DAS-CMOP1 to DAS-CMOP9, and runs each with sixteen triplets (its table 3). + +DAS-CMOP5 (table 2) has two objectives over 30 variables in [0, 1]: + +```text +f₁ = x₁ + g +f₂ = 1 − √x₁ + g +g = 29 + Σⱼ₌₂³⁰ ((xⱼ − 0.5)² − cos(20π(xⱼ − 0.5))) +subject to + sin(20πx₁) − b ≥ 0, b = 2η − 1 (type I) + (e − g)(g − 0.5) ≥ 0, e = 0.5 − ln ζ (type II) + ((f₁ − p_k) cos θ − (f₂ − q_k) sin θ)²/0.3 + + ((f₁ − p_k) sin θ + (f₂ − q_k) cos θ)²/1.2 ≥ r, r = γ/2 (type III, k = 1, …, 9) +θ = −π/4, (p_k, q_k) = (0, 1.5), (1, 0.5), (0, 2.5), (1, 1.5), (2, 0.5), (0, 3.5), (1, 2.5), (2, 1.5), (3, 0.5) +``` + +Both objectives are minimized. At g = 0 the unconstrained front is convex. The distance function g is 0 with every xⱼ at 0.5, and has 11²⁹ − 1 local minima, the cosine's, as DTLZ1's. The example uses the triplet with which the paper's figure 6 plots DAS-CMOP5, (0.5, 0.5, 0.5). Then b = 0: sin(20πx₁) ≥ 0 keeps x₁ in [0, 0.05], [0.1, 0.15], …, half of it; g must be between 0.5 and 0.5 + ln 2 ≈ 1.193, which moves the front out along the diagonal by 0.5; and the nine ellipses, each 45° from the axes, have semi-axes √(0.3 r) ≈ 0.27 along the diagonal and √(1.2 r) ≈ 0.55 across it. + +The front depends on the triplet, and the paper samples it rather than writing it down; genoxide samples it in the same way, as the first feasible point of each of 20,000 rays α(x₁) + g (1, 1) that the type I constraint allows, non-dominated. With this triplet it is ten pieces, f₁ from 0.5 to 0.55, 0.6 to 0.65, 0.7 to 0.75, 0.8 to 0.855, 0.9206 to 0.9793, 1.0328 to 1.0823, 1.1284 to 1.1715, 1.2118 to 1.25, 1.3 to 1.35 and 1.4 to 1.45: the type I intervals of x₁, with f₁ from 0.838 to 1.248 on the boundary of the ellipse centered at (1, 0.5). Its ideal point is (0.5, 0.5253) and its nadir point (1.45, 1.5). The authors published sampled fronts for the sixteen triplets with their code, which genoxide's agree with (the docs of `multi::problems::DasCmop1` say how they were compared). + +## What makes it hard + +The multimodal g first: its local minima put local fronts parallel to the true one, and the type II band, g between 0.5 and e, is itself a target among them: a population that reaches the band at a local minimum of g must leave the band to improve, through infeasible solutions. Then the type I constraint, which cuts the front into short pieces that a population must spread over, and the ellipses, which block the way to some of them: a population that converges behind one stays there, on the far side, with constraint dominance, which never accepts an infeasible step. + +## Representation + +A `Real` genome of 30 genes in [0, 1]. The problem is genoxide's `DasCmop5`, whose fitness is the two objectives and the total constraint violation, 0 when it is feasible; the triplet is a `Difficulty`, `Difficulty::standard(k)` for the paper's sixteen. In Python, `gx.problems.DasCmop5()`, with `difficulty=(η, ζ, γ)` or the number of one of the sixteen, which `run` evaluates in Rust, so both versions print the same. + +Solutions compare by constrained dominance, the rule of the NSGA-II paper and the constraint handling that the paper calls CDP. A feasible solution beats an infeasible one. Of two infeasible ones, the smaller violation wins. Of two feasible ones, Pareto dominance decides. + +## Algorithm + +NSGA-II (Deb, Pratap, Agarwal and Meyarivan, 2002, IEEE Transactions on Evolutionary Computation 6(2): 182-197) with the paper's settings: a population of 300 for 1,000 generations, 300,000 evaluations (section 7.1), simulated binary crossover with η = 20 at a rate of 0.9, and polynomial mutation at a rate of 1/30 per gene, twice: + +- with η = 20 for the mutation, the paper's; +- with η = 5, whose steps are larger: the median step is about 3% of the range with η = 20 and about 11% with η = 5. + +The larger steps of η = 5 keep the population spread over x₁ while it converges, so that it reaches every piece. + +## Output + +A line per run: the size of its final front, how many of its solutions are feasible, and the front's IGD+ and hypervolume. Then the hypervolume of the whole optimal front. + +Both indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 500 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest point of the found front, counting only the objectives in which the found point is worse. Smaller is better, and 0 means the found front covers the optimal one. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the area the front dominates up to the reference point (1.1, 1.1). Larger is better; the whole front's is computed from 20,000 of its points. Only feasible solutions count. + +The page plays both runs back over the grey feasible region, sampled from genomes with x₁ evenly spread and every distance variable at one value, swept; hollow points are infeasible solutions of the populations (at most 100 a frame), and the line is the optimal front, in its pieces. + +[The project page](https://tachsin.gr/projects/genoxide/examples/das-cmop5) plays this run back. + +## Good results + +The target: an IGD+ of at most 0.01 in normalized objectives, with every solution feasible. + +With the paper's settings, the run with seed 1 ends with 300 solutions, 300 feasible, IGD+ 0.0006, hypervolume 0.8482. With η = 5, 300 solutions, 300 feasible, IGD+ 0.0007, hypervolume 0.8478. Over seeds 1 to 20, with η = 20, 18 runs reach the target; the median IGD+ is 0.0006, the worst 0.5659; with η = 5, every run reaches the target, with IGD+ from 0.0006 to 0.0008. + +The paper's NSGA-II-CDP ends with a mean IGD of 0.00102 on DAS-CMOP5 with this triplet (its table 4, number 8), and MOEA/D-CDP with 0.00203. diff --git a/examples/das_cmop5/main.py b/examples/das_cmop5/main.py new file mode 100644 index 00000000..b03f2d73 --- /dev/null +++ b/examples/das_cmop5/main.py @@ -0,0 +1,77 @@ +"""DAS-CMOP5: minimize two objectives over 30 variables subject to 11 constraints, whose front is pieces of a convex curve and of ellipses, +with NSGA-II. + +Fan et al.'s DAS-CMOP5, from genoxide's problems.DasCmop5, with the difficulty triplet of the paper's +figure 6, (0.5, 0.5, 0.5); run evaluates it in Rust. Runs NSGA-II twice with the paper's settings, a +population of 300 for 1,000 generations (300,000 evaluations): with polynomial mutation with +η = 20, as the paper's, and with η = 5. Prints each final front's size, how many of its solutions +are feasible, and its IGD+ to 500 points of the optimal front and hypervolume, with the objectives +normalized by the front's ideal and nadir points; then the whole front's hypervolume. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/das_cmop5/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points: 1.1 times the nadir point +REFERENCE = [1.1, 1.1] + +# the paper's population and evaluations: 300 for 1,000 generations +POPULATION = 300 +GENERATIONS = 1_000 + +problem = gx.problems.DasCmop5() +ideal, nadir = problem.ideal_point, problem.nadir_point + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) + + +def run(name, eta): + """Runs NSGA-II with the paper's population, simulated binary crossover with η = 20 at + genoxide's default rate of 0.9, and polynomial mutation with the distribution index ``eta`` + at a rate of 1/n per gene; prints its final front's size, how many of it are feasible, and its + IGD+ to 500 points of the optimal front and hypervolume, with normalized objectives.""" + nsga2 = gx.Nsga2( + problem.genome, + objectives=problem.objectives, + population_size=POPULATION, + crossover=gx.SimulatedBinaryCrossover(20), + mutation=gx.PolynomialMutation(eta, rate=1 / problem.dimensions), + seed=1, + ) + result = nsga2.run(problem, generations=GENERATIONS, on_generation=trace.fronts(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"NSGA-II, {name}, {GENERATIONS} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(500)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + print(start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}") + + +# the paper's settings: polynomial mutation with η = 20 +run("η = 20", 20) +# polynomial mutation with η = 5, whose steps are larger +run("η = 5", 5) +# the hypervolume of the whole front, from 20,000 of its points +whole = normalized(problem.optimal_front(20_000)) +print(f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}") +trace.write() diff --git a/examples/das_cmop5/main.rs b/examples/das_cmop5/main.rs new file mode 100644 index 00000000..9e9f4e32 --- /dev/null +++ b/examples/das_cmop5/main.rs @@ -0,0 +1,104 @@ +//! DAS-CMOP5: minimize two objectives over 30 variables subject to 11 constraints, whose front is pieces of a convex curve and of ellipses, with +//! NSGA-II. +//! +//! Fan et al.'s DAS-CMOP5, from genoxide's `multi::problems::DasCmop5`, with the difficulty triplet of +//! the paper's figure 6, (0.5, 0.5, 0.5). Runs NSGA-II twice with the paper's settings, a population +//! of 300 for 1,000 generations (300,000 evaluations): with polynomial mutation with η = 20, as +//! the paper's, and with η = 5. Prints each final front's size, how many of its solutions are +//! feasible, and its IGD+ to 500 points of the optimal front and hypervolume, with the objectives +//! normalized by the front's ideal and nadir points; then the whole front's hypervolume. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example das_cmop5 +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{DasCmop5, MultiProblem}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points: 1.1 times the nadir point +const REFERENCE: [f64; 2] = [1.1, 1.1]; + +// the paper's population and evaluations: 300 for 1,000 generations +const POPULATION: usize = 300; +const GENERATIONS: u64 = 1_000; + +fn main() -> Result<()> { + let problem = DasCmop5::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the paper's settings: polynomial mutation with η = 20 + run(&problem, "η = 20", 20.0, &mut trace)?; + // polynomial mutation with η = 5, whose steps are larger + run(&problem, "η = 5", 5.0, &mut trace)?; + // the hypervolume of the whole front, from 20,000 of its points + let whole = normalized(&problem, &problem.optimal_front(20_000).expect("known")); + let volume = hypervolume(&whole, &REFERENCE, &[Minimize; 2]); + println!("the whole front: hypervolume {volume:.4}"); + trace.write(&problem); + Ok(()) +} + +// runs NSGA-II with the paper's population, simulated binary crossover with η = 20 at genoxide's +// default rate of 0.9, and polynomial mutation with the distribution index `eta` at a rate of 1/n +// per gene; prints its final front's size, how many of it are feasible, and its IGD+ to 500 +// points of the optimal front and hypervolume, with normalized objectives +fn run(problem: &DasCmop5, name: &'static str, eta: f64, trace: &mut trace::Trace) -> Result<()> { + let rate = 1.0 / problem.variables() as f64; + let nsga2 = Nsga2::builder(problem.representation(), [Minimize; 2]) + .population_size(POPULATION) + .crossover(SimulatedBinaryCrossover::new(20.0)?) + .mutate(PolynomialMutation::per_gene(rate, eta)?) + .seed(1) + .build()?; + let mut record = trace.fronts(name); + let outcome = MultiEngine::new(nsga2, *problem) + .stop_when(Stop::generations(GENERATIONS)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 2], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 2]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "NSGA-II, {name}, {GENERATIONS} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(500).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 2]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 2]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &DasCmop5, points: &[[f64; 2]]) -> Vec<[f64; 2]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 2]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/das_cmop5/output.txt b/examples/das_cmop5/output.txt new file mode 100644 index 00000000..162a0663 --- /dev/null +++ b/examples/das_cmop5/output.txt @@ -0,0 +1,3 @@ +NSGA-II, η = 20, 1000 generations: 300 solutions, 300 feasible, IGD+ 0.0006, hypervolume 0.8482 +NSGA-II, η = 5, 1000 generations: 300 solutions, 300 feasible, IGD+ 0.0007, hypervolume 0.8478 +the whole front: hypervolume 0.8494 diff --git a/examples/das_cmop5/trace.json b/examples/das_cmop5/trace.json new file mode 100644 index 00000000..31346bc3 --- /dev/null +++ b/examples/das_cmop5/trace.json @@ -0,0 +1,42 @@ 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= 20":0.840388,"η = 5":0.84587},"infeasible":{"η = 20":[],"η = 5":[]}}} +]} diff --git a/examples/das_cmop5/trace.py b/examples/das_cmop5/trace.py new file mode 100644 index 00000000..eb0b99fe --- /dev/null +++ b/examples/das_cmop5/trace.py @@ -0,0 +1,260 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: both fronts, each population's infeasible solutions (at most 100) and +feasible share, and the fronts' hypervolumes with normalized objectives, in at most 40 +generations; and the optimal front, in its pieces, and the feasible region, sampled. A frame's +evaluations are the two runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the samples of the feasible region along x₁ and along the distance variables, its grid, and the +# most infeasible solutions a frame shows +SAMPLES = 400 +DISTANCES = 800 +CELLS = 80 +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions, the hypervolume and the feasible share + self.series = {} + + def fronts(self, name): + """The callback for ``run`` that records the run ``name`` after each generation: None + without a trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + feasible = progress.front_violations == 0 + front = progress.front_objectives[feasible] + infeasible = progress.objectives[progress.violations != 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + share = (progress.violations == 0).sum() / len(progress.violations) + history.append( + (progress.evaluations, front.tolist(), infeasible.tolist(), volume, share) + ) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(40) + for generation in range(max(map(len, self.series.values()), default=0)): + at = {name: runs[min(generation, len(runs) - 1)] for name, runs in self.series.items()} + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at.values()), + "best": None, + "median": None, + "state": { + "fronts": {name: record[1] for name, record in at.items()}, + "infeasible": {name: record[2] for name, record in at.items()}, + "hypervolume": {name: record[3] for name, record in at.items()}, + "feasible": {name: record[4] for name, record in at.items()}, + }, + } + ) + whole = self.normalized(self.problem.optimal_front(20_000)) + settings = { + "format": 1, + "example": "das_cmop5", + "objective": ["minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "front-2d", + "problem": { + "objectives": ["f1", "f2"], + "true_front": pieces(self.problem.optimal_front(400).tolist()), + "series": list(self.series), + "feasible_region": region(self.problem), + }, + } + write(self.path, settings, frames.to_list()) + + +def pieces(front): + """The front in pieces, split where two points are more than 2% of its extent apart, each + point once.""" + first, last = front[0], front[-1] + extent = math.hypot(last[0] - first[0], first[1] - last[1]) + split = [] + for point in front: + if not split: + split.append([point]) + continue + previous = split[-1][-1] + if math.hypot(point[0] - previous[0], point[1] - previous[1]) > 0.02 * extent: + split.append([point]) + elif point != previous: + split[-1].append(point) + return split + + +def region(problem): + """The feasible region in objective space: genomes with x₁ evenly spread over [0, 1] and + every distance variable at a value v swept over [0.5, 0.52], which moves g from 0 up; a cell of a + grid of 80 × 80 over [0, 1.25 f₁ᵐᵃˣ] × [0, 1.25 f₂ᵐᵃˣ] (the nadir point's) is feasible if a + feasible genome lands in it, and the rows go up from f₂ = 0.""" + n = problem.dimensions + nadir = problem.nadir_point + width, height = 1.25 * nadir[0], 1.25 * nadir[1] + cells = [[False] * CELLS for _ in range(CELLS)] + for a in range(SAMPLES): + genomes = np.empty((DISTANCES, n)) + genomes[:, 0] = a / (SAMPLES - 1) + for b in range(DISTANCES): + genomes[b, 1:] = 0.5 + 0.02 * b / (DISTANCES - 1) + objectives, violations = problem.evaluate(genomes) + i = np.floor(objectives[:, 0] / width * CELLS) + j = np.floor(objectives[:, 1] / height * CELLS) + inside = (violations == 0) & (i >= 0) & (j >= 0) & (i < CELLS) & (j < CELLS) + for column, row in zip(i[inside].astype(int), j[inside].astype(int)): + cells[row][column] = True + # each row as the pairs of columns [start, end) where it's feasible + rows = [runs(row) for row in cells] + return {"x": [0.0, width], "y": [0.0, height], "columns": CELLS, "rows": rows} + + +def runs(row): + """The columns where a row of cells turns feasible or infeasible, as pairs [start, end).""" + turns = [ + column for column in range(len(row)) if row[column] != (column > 0 and row[column - 1]) + ] + return turns + ([len(row)] if row[-1] else []) + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/das_cmop5/trace.rs b/examples/das_cmop5/trace.rs new file mode 100644 index 00000000..65cf63f9 --- /dev/null +++ b/examples/das_cmop5/trace.rs @@ -0,0 +1,367 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: both fronts, each population's infeasible solutions (at most 100) and +//! feasible share, and the fronts' hypervolumes with normalized objectives, in at most 40 +//! generations; and the optimal front, in its pieces, and the feasible region, sampled. A frame's +//! evaluations are the two runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{DasCmop5, MultiProblem}; +use genoxide::multi::{MultiFitnessFunction, MultiSnapshot, Scores}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, the +// hypervolume and the feasible share +type Record = (u64, Vec<[f64; 2]>, Vec<[f64; 2]>, f64, f64); + +// the samples of the feasible region along x₁ and along the distance variables, its grid, and the +// most infeasible solutions a frame shows +const SAMPLES: usize = 400; +const DISTANCES: usize = 800; +const CELLS: usize = 80; +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation + pub fn fronts(&mut self, name: &'static str) -> impl FnMut(&MultiSnapshot<'_, Reals, 2>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let values = |x: &Individual>| x.fitness()?.values(); + let feasible = |x: &&Individual>| { + x.fitness().is_some_and(|scores| scores.is_feasible()) + }; + let front: Vec<[f64; 2]> = snapshot + .front() + .iter() + .filter(feasible) + .filter_map(values) + .collect(); + let population = snapshot.population().as_slice(); + let infeasible = population.iter().filter(|x| !feasible(x)); + let infeasible: Vec<[f64; 2]> = infeasible.filter_map(values).take(SHOWN).collect(); + let share = population.iter().filter(feasible).count() as f64 / population.len() as f64; + let volume = hypervolume( + &normalized(&DasCmop5::default(), &front), + &REFERENCE, + &[Minimize; 2], + ); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume, share)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &DasCmop5) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(40); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let mut state = [Map::new(), Map::new(), Map::new(), Map::new()]; + let mut evaluations = 0; + for (name, history) in &self.series { + let (done, front, infeasible, volume, share) = + &history[generation.min(history.len() - 1)]; + let values = [json!(front), json!(infeasible), json!(volume), json!(share)]; + for (map, value) in state.iter_mut().zip(values) { + map.insert(name.to_string(), value); + } + evaluations += done; + } + let [fronts, infeasible, volumes, shares] = state; + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "state": { + "fronts": fronts, + "infeasible": infeasible, + "hypervolume": volumes, + "feasible": shares, + }, + })); + } + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(20_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "das_cmop5", + "objective": ["minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 2]), + "plot": "front-2d", + "problem": { + "objectives": ["f1", "f2"], + "true_front": pieces(&problem.optimal_front(400).expect("known")), + "series": names, + "feasible_region": region(problem), + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the front in pieces, split where two points are more than 2% of its extent apart, each point +// once +fn pieces(front: &[[f64; 2]]) -> Vec> { + let (first, last) = (front[0], front[front.len() - 1]); + let extent = (last[0] - first[0]).hypot(first[1] - last[1]); + let mut pieces: Vec> = Vec::new(); + for &point in front { + match pieces.last_mut() { + Some(piece) => { + let previous = piece[piece.len() - 1]; + if (point[0] - previous[0]).hypot(point[1] - previous[1]) > 0.02 * extent { + pieces.push(vec![point]); + } else if point != previous { + piece.push(point); + } + } + None => pieces.push(vec![point]), + } + } + pieces +} + +// the feasible region in objective space: genomes with x₁ evenly spread over [0, 1] and every +// distance variable at a value v swept over [0.5, 0.52], which moves g from 0 up; a cell of a grid +// of 80 × 80 over [0, 1.25 f₁ᵐᵃˣ] × [0, 1.25 f₂ᵐᵃˣ] (the nadir point's) is feasible if a feasible +// genome lands in it, and the rows go up from f₂ = 0 +fn region(problem: &DasCmop5) -> Value { + let n = problem.variables(); + let nadir = problem.nadir_point().expect("known"); + let (width, height) = (1.25 * nadir[0], 1.25 * nadir[1]); + let mut cells = vec![vec![false; CELLS]; CELLS]; + for a in 0..SAMPLES { + let mut x = vec![a as f64 / (SAMPLES - 1) as f64; 1]; + x.resize(n, 0.0); + for b in 0..DISTANCES { + let v = 0.5 + 0.02 * b as f64 / (DISTANCES - 1) as f64; + x[1..].fill(v); + let (f, violation) = problem.evaluate(&Reals::from(x.clone())); + let i = (f[0] / width * CELLS as f64).floor(); + let j = (f[1] / height * CELLS as f64).floor(); + if violation == 0.0 && i >= 0.0 && j >= 0.0 && i < CELLS as f64 && j < CELLS as f64 { + cells[j as usize][i as usize] = true; + } + } + } + // each row as the pairs of columns [start, end) where it's feasible + let rows: Vec> = cells.iter().map(Vec::as_slice).map(runs).collect(); + json!({"x": [0.0, width], "y": [0.0, height], "columns": CELLS, "rows": rows}) +} + +// the columns where a row of cells turns feasible or infeasible, as pairs [start, end) +fn runs(row: &[bool]) -> Vec { + let mut runs = Vec::new(); + for (column, &inside) in row.iter().enumerate() { + if inside != (column > 0 && row[column - 1]) { + runs.push(column); + } + } + if row.last() == Some(&true) { + runs.push(row.len()); + } + runs +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/das_cmop6/README.md b/examples/das_cmop6/README.md new file mode 100644 index 00000000..092533d0 --- /dev/null +++ b/examples/das_cmop6/README.md @@ -0,0 +1,72 @@ +--- +title: DAS-CMOP6 +category: multi-objective +summary: Minimize two objectives over 30 variables subject to 11 constraints of adjustable difficulty, whose Pareto front is disconnected pieces, with NSGA-II. +reference: "Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). Difficulty adjustable and scalable constrained multiobjective test problem toolkit. Evolutionary Computation 28(3): 339-378." +reference_url: https://doi.org/10.1162/evco_a_00259 +optimum: "for the difficulty triplet (0.5, 0.5, 0.5), six pieces, some of them points; ideal point (0.5, 0.6056), nadir point (1.3, 1.5); hypervolume 0.7211 (normalized objectives, reference point (1.1, 1.1))" +languages: [rust, python] +order: 212 +family: DAS-CMOP +--- + +# DAS-CMOP6 + +## The problem + +Fan et al. (2020) built a toolkit of constrained test problems whose difficulty is set by three numbers, a triplet (η, ζ, γ) in [0, 1]³, one for each kind of difficulty: η for diversity, ζ for feasibility and γ for convergence. Each kind comes from a type of constraint. Type I constraints cut the front into pieces, narrower as η grows. A type II constraint asks the distance from the unconstrained front to be in a band, narrower as ζ grows. Type III constraints are infeasible regions in the objective space that block the way to the front, larger as γ grows. The paper suggests nine problems built this way, DAS-CMOP1 to DAS-CMOP9, and runs each with sixteen triplets (its table 3). + +DAS-CMOP6 (table 2) has two objectives over 30 variables in [0, 1]: + +```text +f₁ = x₁ + g +f₂ = 1 − √x₁ + 0.5 |sin(5πx₁)| + g +g = 29 + Σⱼ₌₂³⁰ ((xⱼ − 0.5)² − cos(20π(xⱼ − 0.5))) +subject to + sin(20πx₁) − b ≥ 0, b = 2η − 1 (type I) + (e − g)(g − 0.5) ≥ 0, e = 0.5 − ln ζ (type II) + ((f₁ − p_k) cos θ − (f₂ − q_k) sin θ)²/0.3 + + ((f₁ − p_k) sin θ + (f₂ − q_k) cos θ)²/1.2 ≥ r, r = γ/2 (type III, k = 1, …, 9) +θ = −π/4, (p_k, q_k) = (0, 1.5), (1, 0.5), (0, 2.5), (1, 1.5), (2, 0.5), (0, 3.5), (1, 2.5), (2, 1.5), (3, 0.5) +``` + +Both objectives are minimized. At g = 0 the unconstrained front is disconnected. The distance function g is 0 with every xⱼ at 0.5, and has 11²⁹ − 1 local minima, the cosine's, as DTLZ1's. The example uses the triplet with which the paper's figure 6 plots DAS-CMOP6, (0.5, 0.5, 0.5). Then b = 0: sin(20πx₁) ≥ 0 keeps x₁ in [0, 0.05], [0.1, 0.15], …, half of it; g must be between 0.5 and 0.5 + ln 2 ≈ 1.193, which moves the front out along the diagonal by 0.5; and the nine ellipses, each 45° from the axes, have semi-axes √(0.3 r) ≈ 0.27 along the diagonal and √(1.2 r) ≈ 0.55 across it. + +The front depends on the triplet, and the paper samples it rather than writing it down; genoxide samples it in the same way, as the first feasible point of each of 20,000 rays α(x₁) + g (1, 1) that the type I constraint allows, non-dominated. With this triplet it is six pieces, some of them single points: f₁ from 0.5 to 0.504, 0.6498 to 0.65, 0.7, 0.904 to 0.9206, 1.1065 to 1.1284, and 1.3, those from 0.904 to 1.128 on the boundary of the ellipse centered at (1, 0.5): the type I intervals of x₁ and the bumps of the shape leave little. Its ideal point is (0.5, 0.6056) and its nadir point (1.3, 1.5). The authors published sampled fronts for the sixteen triplets with their code, which genoxide's agree with (the docs of `multi::problems::DasCmop1` say how they were compared). + +## What makes it hard + +The multimodal g first: its local minima put local fronts parallel to the true one, and the type II band, g between 0.5 and e, is itself a target among them: a population that reaches the band at a local minimum of g must leave the band to improve, through infeasible solutions. Then the type I constraint, which cuts the front into short pieces that a population must spread over, and the ellipses, which block the way to some of them: a population that converges behind one stays there, on the far side, with constraint dominance, which never accepts an infeasible step. + +## Representation + +A `Real` genome of 30 genes in [0, 1]. The problem is genoxide's `DasCmop6`, whose fitness is the two objectives and the total constraint violation, 0 when it is feasible; the triplet is a `Difficulty`, `Difficulty::standard(k)` for the paper's sixteen. In Python, `gx.problems.DasCmop6()`, with `difficulty=(η, ζ, γ)` or the number of one of the sixteen, which `run` evaluates in Rust, so both versions print the same. + +Solutions compare by constrained dominance, the rule of the NSGA-II paper and the constraint handling that the paper calls CDP. A feasible solution beats an infeasible one. Of two infeasible ones, the smaller violation wins. Of two feasible ones, Pareto dominance decides. + +## Algorithm + +NSGA-II (Deb, Pratap, Agarwal and Meyarivan, 2002, IEEE Transactions on Evolutionary Computation 6(2): 182-197) with the paper's settings: a population of 300 for 1,000 generations, 300,000 evaluations (section 7.1), simulated binary crossover with η = 20 at a rate of 0.9, and polynomial mutation at a rate of 1/30 per gene, twice: + +- with η = 20 for the mutation, the paper's; +- with η = 5, whose steps are larger: the median step is about 3% of the range with η = 20 and about 11% with η = 5. + +The larger steps of η = 5 keep the population spread over x₁ while it converges, so that it reaches every piece. + +## Output + +A line per run: the size of its final front, how many of its solutions are feasible, and the front's IGD+ and hypervolume. Then the hypervolume of the whole optimal front. + +Both indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 500 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest point of the found front, counting only the objectives in which the found point is worse. Smaller is better, and 0 means the found front covers the optimal one. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the area the front dominates up to the reference point (1.1, 1.1). Larger is better; the whole front's is computed from 20,000 of its points. Only feasible solutions count. + +The page plays both runs back over the grey feasible region, sampled from genomes with x₁ evenly spread and every distance variable at one value, swept; hollow points are infeasible solutions of the populations (at most 100 a frame), and the line is the optimal front, in its pieces. + +[The project page](https://tachsin.gr/projects/genoxide/examples/das-cmop6) plays this run back. + +## Good results + +The target: an IGD+ of at most 0.01 in normalized objectives, with every solution feasible. + +With the paper's settings, the run with seed 1 ends with 28 solutions, 28 feasible, IGD+ 0.0511, hypervolume 0.6538. With η = 5, 300 solutions, 300 feasible, IGD+ 0.0008, hypervolume 0.7202. Over seeds 1 to 20, with η = 20, 7 runs reach the target, one ends with no feasible solution, and the median IGD+ is 0.0149; with η = 5, 19 runs reach the target, with IGD+ from 0.0004 to 0.0010, and one ends at 0.0915, short of a piece. + +The paper's NSGA-II-CDP ends with a mean IGD of 0.190 on DAS-CMOP6 with this triplet (its table 5, number 8), with a standard deviation of 0.290, and MOEA/D-CDP with 0.650. diff --git a/examples/das_cmop6/main.py b/examples/das_cmop6/main.py new file mode 100644 index 00000000..69907056 --- /dev/null +++ b/examples/das_cmop6/main.py @@ -0,0 +1,77 @@ +"""DAS-CMOP6: minimize two objectives over 30 variables subject to 11 constraints, whose front is disconnected, +with NSGA-II. + +Fan et al.'s DAS-CMOP6, from genoxide's problems.DasCmop6, with the difficulty triplet of the paper's +figure 6, (0.5, 0.5, 0.5); run evaluates it in Rust. Runs NSGA-II twice with the paper's settings, a +population of 300 for 1,000 generations (300,000 evaluations): with polynomial mutation with +η = 20, as the paper's, and with η = 5. Prints each final front's size, how many of its solutions +are feasible, and its IGD+ to 500 points of the optimal front and hypervolume, with the objectives +normalized by the front's ideal and nadir points; then the whole front's hypervolume. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/das_cmop6/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points: 1.1 times the nadir point +REFERENCE = [1.1, 1.1] + +# the paper's population and evaluations: 300 for 1,000 generations +POPULATION = 300 +GENERATIONS = 1_000 + +problem = gx.problems.DasCmop6() +ideal, nadir = problem.ideal_point, problem.nadir_point + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) + + +def run(name, eta): + """Runs NSGA-II with the paper's population, simulated binary crossover with η = 20 at + genoxide's default rate of 0.9, and polynomial mutation with the distribution index ``eta`` + at a rate of 1/n per gene; prints its final front's size, how many of it are feasible, and its + IGD+ to 500 points of the optimal front and hypervolume, with normalized objectives.""" + nsga2 = gx.Nsga2( + problem.genome, + objectives=problem.objectives, + population_size=POPULATION, + crossover=gx.SimulatedBinaryCrossover(20), + mutation=gx.PolynomialMutation(eta, rate=1 / problem.dimensions), + seed=1, + ) + result = nsga2.run(problem, generations=GENERATIONS, on_generation=trace.fronts(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"NSGA-II, {name}, {GENERATIONS} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(500)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + print(start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}") + + +# the paper's settings: polynomial mutation with η = 20 +run("η = 20", 20) +# polynomial mutation with η = 5, whose steps are larger +run("η = 5", 5) +# the hypervolume of the whole front, from 20,000 of its points +whole = normalized(problem.optimal_front(20_000)) +print(f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}") +trace.write() diff --git a/examples/das_cmop6/main.rs b/examples/das_cmop6/main.rs new file mode 100644 index 00000000..4a70f546 --- /dev/null +++ b/examples/das_cmop6/main.rs @@ -0,0 +1,104 @@ +//! DAS-CMOP6: minimize two objectives over 30 variables subject to 11 constraints, whose front is disconnected, with +//! NSGA-II. +//! +//! Fan et al.'s DAS-CMOP6, from genoxide's `multi::problems::DasCmop6`, with the difficulty triplet of +//! the paper's figure 6, (0.5, 0.5, 0.5). Runs NSGA-II twice with the paper's settings, a population +//! of 300 for 1,000 generations (300,000 evaluations): with polynomial mutation with η = 20, as +//! the paper's, and with η = 5. Prints each final front's size, how many of its solutions are +//! feasible, and its IGD+ to 500 points of the optimal front and hypervolume, with the objectives +//! normalized by the front's ideal and nadir points; then the whole front's hypervolume. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example das_cmop6 +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{DasCmop6, MultiProblem}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points: 1.1 times the nadir point +const REFERENCE: [f64; 2] = [1.1, 1.1]; + +// the paper's population and evaluations: 300 for 1,000 generations +const POPULATION: usize = 300; +const GENERATIONS: u64 = 1_000; + +fn main() -> Result<()> { + let problem = DasCmop6::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the paper's settings: polynomial mutation with η = 20 + run(&problem, "η = 20", 20.0, &mut trace)?; + // polynomial mutation with η = 5, whose steps are larger + run(&problem, "η = 5", 5.0, &mut trace)?; + // the hypervolume of the whole front, from 20,000 of its points + let whole = normalized(&problem, &problem.optimal_front(20_000).expect("known")); + let volume = hypervolume(&whole, &REFERENCE, &[Minimize; 2]); + println!("the whole front: hypervolume {volume:.4}"); + trace.write(&problem); + Ok(()) +} + +// runs NSGA-II with the paper's population, simulated binary crossover with η = 20 at genoxide's +// default rate of 0.9, and polynomial mutation with the distribution index `eta` at a rate of 1/n +// per gene; prints its final front's size, how many of it are feasible, and its IGD+ to 500 +// points of the optimal front and hypervolume, with normalized objectives +fn run(problem: &DasCmop6, name: &'static str, eta: f64, trace: &mut trace::Trace) -> Result<()> { + let rate = 1.0 / problem.variables() as f64; + let nsga2 = Nsga2::builder(problem.representation(), [Minimize; 2]) + .population_size(POPULATION) + .crossover(SimulatedBinaryCrossover::new(20.0)?) + .mutate(PolynomialMutation::per_gene(rate, eta)?) + .seed(1) + .build()?; + let mut record = trace.fronts(name); + let outcome = MultiEngine::new(nsga2, *problem) + .stop_when(Stop::generations(GENERATIONS)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 2], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 2]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "NSGA-II, {name}, {GENERATIONS} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(500).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 2]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 2]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &DasCmop6, points: &[[f64; 2]]) -> Vec<[f64; 2]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 2]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/das_cmop6/output.txt b/examples/das_cmop6/output.txt new file mode 100644 index 00000000..27507bab --- /dev/null +++ b/examples/das_cmop6/output.txt @@ -0,0 +1,3 @@ +NSGA-II, η = 20, 1000 generations: 28 solutions, 28 feasible, IGD+ 0.0511, hypervolume 0.6538 +NSGA-II, η = 5, 1000 generations: 300 solutions, 300 feasible, IGD+ 0.0008, hypervolume 0.7202 +the whole front: hypervolume 0.7211 diff --git a/examples/das_cmop6/trace.json b/examples/das_cmop6/trace.json new file mode 100644 index 00000000..06211b7c --- /dev/null +++ b/examples/das_cmop6/trace.json @@ -0,0 +1,42 @@ 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= 20":0.647664,"η = 5":0.71971},"infeasible":{"η = 20":[],"η = 5":[]}}} +]} diff --git a/examples/das_cmop6/trace.py b/examples/das_cmop6/trace.py new file mode 100644 index 00000000..cafd5f68 --- /dev/null +++ b/examples/das_cmop6/trace.py @@ -0,0 +1,260 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: both fronts, each population's infeasible solutions (at most 100) and +feasible share, and the fronts' hypervolumes with normalized objectives, in at most 40 +generations; and the optimal front, in its pieces, and the feasible region, sampled. A frame's +evaluations are the two runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the samples of the feasible region along x₁ and along the distance variables, its grid, and the +# most infeasible solutions a frame shows +SAMPLES = 400 +DISTANCES = 800 +CELLS = 80 +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions, the hypervolume and the feasible share + self.series = {} + + def fronts(self, name): + """The callback for ``run`` that records the run ``name`` after each generation: None + without a trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + feasible = progress.front_violations == 0 + front = progress.front_objectives[feasible] + infeasible = progress.objectives[progress.violations != 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + share = (progress.violations == 0).sum() / len(progress.violations) + history.append( + (progress.evaluations, front.tolist(), infeasible.tolist(), volume, share) + ) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(40) + for generation in range(max(map(len, self.series.values()), default=0)): + at = {name: runs[min(generation, len(runs) - 1)] for name, runs in self.series.items()} + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at.values()), + "best": None, + "median": None, + "state": { + "fronts": {name: record[1] for name, record in at.items()}, + "infeasible": {name: record[2] for name, record in at.items()}, + "hypervolume": {name: record[3] for name, record in at.items()}, + "feasible": {name: record[4] for name, record in at.items()}, + }, + } + ) + whole = self.normalized(self.problem.optimal_front(20_000)) + settings = { + "format": 1, + "example": "das_cmop6", + "objective": ["minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "front-2d", + "problem": { + "objectives": ["f1", "f2"], + "true_front": pieces(self.problem.optimal_front(400).tolist()), + "series": list(self.series), + "feasible_region": region(self.problem), + }, + } + write(self.path, settings, frames.to_list()) + + +def pieces(front): + """The front in pieces, split where two points are more than 2% of its extent apart, each + point once.""" + first, last = front[0], front[-1] + extent = math.hypot(last[0] - first[0], first[1] - last[1]) + split = [] + for point in front: + if not split: + split.append([point]) + continue + previous = split[-1][-1] + if math.hypot(point[0] - previous[0], point[1] - previous[1]) > 0.02 * extent: + split.append([point]) + elif point != previous: + split[-1].append(point) + return split + + +def region(problem): + """The feasible region in objective space: genomes with x₁ evenly spread over [0, 1] and + every distance variable at a value v swept over [0.5, 0.52], which moves g from 0 up; a cell of a + grid of 80 × 80 over [0, 1.25 f₁ᵐᵃˣ] × [0, 1.25 f₂ᵐᵃˣ] (the nadir point's) is feasible if a + feasible genome lands in it, and the rows go up from f₂ = 0.""" + n = problem.dimensions + nadir = problem.nadir_point + width, height = 1.25 * nadir[0], 1.25 * nadir[1] + cells = [[False] * CELLS for _ in range(CELLS)] + for a in range(SAMPLES): + genomes = np.empty((DISTANCES, n)) + genomes[:, 0] = a / (SAMPLES - 1) + for b in range(DISTANCES): + genomes[b, 1:] = 0.5 + 0.02 * b / (DISTANCES - 1) + objectives, violations = problem.evaluate(genomes) + i = np.floor(objectives[:, 0] / width * CELLS) + j = np.floor(objectives[:, 1] / height * CELLS) + inside = (violations == 0) & (i >= 0) & (j >= 0) & (i < CELLS) & (j < CELLS) + for column, row in zip(i[inside].astype(int), j[inside].astype(int)): + cells[row][column] = True + # each row as the pairs of columns [start, end) where it's feasible + rows = [runs(row) for row in cells] + return {"x": [0.0, width], "y": [0.0, height], "columns": CELLS, "rows": rows} + + +def runs(row): + """The columns where a row of cells turns feasible or infeasible, as pairs [start, end).""" + turns = [ + column for column in range(len(row)) if row[column] != (column > 0 and row[column - 1]) + ] + return turns + ([len(row)] if row[-1] else []) + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/das_cmop6/trace.rs b/examples/das_cmop6/trace.rs new file mode 100644 index 00000000..8947a982 --- /dev/null +++ b/examples/das_cmop6/trace.rs @@ -0,0 +1,367 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: both fronts, each population's infeasible solutions (at most 100) and +//! feasible share, and the fronts' hypervolumes with normalized objectives, in at most 40 +//! generations; and the optimal front, in its pieces, and the feasible region, sampled. A frame's +//! evaluations are the two runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{DasCmop6, MultiProblem}; +use genoxide::multi::{MultiFitnessFunction, MultiSnapshot, Scores}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, the +// hypervolume and the feasible share +type Record = (u64, Vec<[f64; 2]>, Vec<[f64; 2]>, f64, f64); + +// the samples of the feasible region along x₁ and along the distance variables, its grid, and the +// most infeasible solutions a frame shows +const SAMPLES: usize = 400; +const DISTANCES: usize = 800; +const CELLS: usize = 80; +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation + pub fn fronts(&mut self, name: &'static str) -> impl FnMut(&MultiSnapshot<'_, Reals, 2>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let values = |x: &Individual>| x.fitness()?.values(); + let feasible = |x: &&Individual>| { + x.fitness().is_some_and(|scores| scores.is_feasible()) + }; + let front: Vec<[f64; 2]> = snapshot + .front() + .iter() + .filter(feasible) + .filter_map(values) + .collect(); + let population = snapshot.population().as_slice(); + let infeasible = population.iter().filter(|x| !feasible(x)); + let infeasible: Vec<[f64; 2]> = infeasible.filter_map(values).take(SHOWN).collect(); + let share = population.iter().filter(feasible).count() as f64 / population.len() as f64; + let volume = hypervolume( + &normalized(&DasCmop6::default(), &front), + &REFERENCE, + &[Minimize; 2], + ); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume, share)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &DasCmop6) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(40); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let mut state = [Map::new(), Map::new(), Map::new(), Map::new()]; + let mut evaluations = 0; + for (name, history) in &self.series { + let (done, front, infeasible, volume, share) = + &history[generation.min(history.len() - 1)]; + let values = [json!(front), json!(infeasible), json!(volume), json!(share)]; + for (map, value) in state.iter_mut().zip(values) { + map.insert(name.to_string(), value); + } + evaluations += done; + } + let [fronts, infeasible, volumes, shares] = state; + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "state": { + "fronts": fronts, + "infeasible": infeasible, + "hypervolume": volumes, + "feasible": shares, + }, + })); + } + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(20_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "das_cmop6", + "objective": ["minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 2]), + "plot": "front-2d", + "problem": { + "objectives": ["f1", "f2"], + "true_front": pieces(&problem.optimal_front(400).expect("known")), + "series": names, + "feasible_region": region(problem), + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the front in pieces, split where two points are more than 2% of its extent apart, each point +// once +fn pieces(front: &[[f64; 2]]) -> Vec> { + let (first, last) = (front[0], front[front.len() - 1]); + let extent = (last[0] - first[0]).hypot(first[1] - last[1]); + let mut pieces: Vec> = Vec::new(); + for &point in front { + match pieces.last_mut() { + Some(piece) => { + let previous = piece[piece.len() - 1]; + if (point[0] - previous[0]).hypot(point[1] - previous[1]) > 0.02 * extent { + pieces.push(vec![point]); + } else if point != previous { + piece.push(point); + } + } + None => pieces.push(vec![point]), + } + } + pieces +} + +// the feasible region in objective space: genomes with x₁ evenly spread over [0, 1] and every +// distance variable at a value v swept over [0.5, 0.52], which moves g from 0 up; a cell of a grid +// of 80 × 80 over [0, 1.25 f₁ᵐᵃˣ] × [0, 1.25 f₂ᵐᵃˣ] (the nadir point's) is feasible if a feasible +// genome lands in it, and the rows go up from f₂ = 0 +fn region(problem: &DasCmop6) -> Value { + let n = problem.variables(); + let nadir = problem.nadir_point().expect("known"); + let (width, height) = (1.25 * nadir[0], 1.25 * nadir[1]); + let mut cells = vec![vec![false; CELLS]; CELLS]; + for a in 0..SAMPLES { + let mut x = vec![a as f64 / (SAMPLES - 1) as f64; 1]; + x.resize(n, 0.0); + for b in 0..DISTANCES { + let v = 0.5 + 0.02 * b as f64 / (DISTANCES - 1) as f64; + x[1..].fill(v); + let (f, violation) = problem.evaluate(&Reals::from(x.clone())); + let i = (f[0] / width * CELLS as f64).floor(); + let j = (f[1] / height * CELLS as f64).floor(); + if violation == 0.0 && i >= 0.0 && j >= 0.0 && i < CELLS as f64 && j < CELLS as f64 { + cells[j as usize][i as usize] = true; + } + } + } + // each row as the pairs of columns [start, end) where it's feasible + let rows: Vec> = cells.iter().map(Vec::as_slice).map(runs).collect(); + json!({"x": [0.0, width], "y": [0.0, height], "columns": CELLS, "rows": rows}) +} + +// the columns where a row of cells turns feasible or infeasible, as pairs [start, end) +fn runs(row: &[bool]) -> Vec { + let mut runs = Vec::new(); + for (column, &inside) in row.iter().enumerate() { + if inside != (column > 0 && row[column - 1]) { + runs.push(column); + } + } + if row.last() == Some(&true) { + runs.push(row.len()); + } + runs +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/das_cmop7/README.md b/examples/das_cmop7/README.md new file mode 100644 index 00000000..05703781 --- /dev/null +++ b/examples/das_cmop7/README.md @@ -0,0 +1,68 @@ +--- +title: DAS-CMOP7 +category: multi-objective +summary: Minimize three objectives over 30 variables subject to 7 constraints of adjustable difficulty, whose Pareto front is patches of the plane f₁ + f₂ + f₃ = 2.5, with NSGA-II and NSGA-III. +reference: "Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). Difficulty adjustable and scalable constrained multiobjective test problem toolkit. Evolutionary Computation 28(3): 339-378." +reference_url: https://doi.org/10.1162/evco_a_00259 +optimum: "for the difficulty triplet (0.5, 0.5, 0.5), patches of the plane f₁ + f₂ + f₃ = 2.5; ideal point (0.5, 0.5, 0.5), nadir point (1.4479, 1.5, 1.5); hypervolume 1.1300 (normalized objectives, reference point (1.1, 1.1, 1.1))" +languages: [rust, python] +order: 213 +family: DAS-CMOP +--- + +# DAS-CMOP7 + +## The problem + +Fan et al. (2020) built a toolkit of constrained test problems whose difficulty is set by a triplet (η, ζ, γ) in [0, 1]³: η for diversity (type I constraints, which cut the front into pieces), ζ for feasibility (a type II constraint, a band of distances from the unconstrained front) and γ for convergence (type III constraints, infeasible regions in the objective space). DAS-CMOP7 to DAS-CMOP9 have three objectives; see [DAS-CMOP1](../das_cmop1/) for the two-objective problems. + +DAS-CMOP7 (table 2) has three objectives over 30 variables in [0, 1]: + +```text +f₁ = x₁x₂ + g +f₂ = x₂(1 − x₁) + g +f₃ = 1 − x₂ + g +g = 28 + Σⱼ₌₃³⁰ ((xⱼ − 0.5)² − cos(20π(xⱼ − 0.5))) +subject to + sin(20πx₁) − b ≥ 0, cos(20πx₂) − b ≥ 0, b = 2η − 1 (type I) + (e − g)(g − 0.5) ≥ 0, e = 0.5 − ln ζ (type II) + Σⱼ fⱼ² − fₖ² + (fₖ − 1)² − r² ≥ 0, k = 1, 2, 3 (type III) + Σⱼ (fⱼ − 1/√3)² − r² ≥ 0, r = γ/2 (type III) +``` + +All three are minimized. At g = 0 the unconstrained front is the simplex f₁ + f₂ + f₃ = 1. The distance function g is 0 with every xⱼ at 0.5, and has 11²⁸ − 1 local minima, the cosine's, as DTLZ1's. The example uses the triplet with which the paper's figure 6 plots DAS-CMOP7, (0.5, 0.5, 0.5): b = 0, which keeps x₁ in [0, 0.05], [0.1, 0.15], … and x₂ in [0, 0.025], [0.075, 0.125], …; g between 0.5 and 0.5 + ln 2 ≈ 1.193; and spheres of radius 0.25. + +The front depends on the triplet, and the paper samples it; genoxide samples it in the same way, as the first feasible point of each ray α(x₁, x₂) + g (1, 1, 1) from the 41,905 points of Das and Dennis's method with 288 divisions, non-dominated. With this triplet it is patches of the plane f₁ + f₂ + f₃ = 2.5, the simplex moved out by 0.5 along the diagonal, where x₁ and x₂ are both in the type I intervals. Its ideal point is (0.5, 0.5, 0.5) and its nadir point (1.4479, 1.5, 1.5) (1.4479, short of 1.45, where the sampled rays end). + +## What makes it hard + +The grid of patches first: the type I constraints keep x₁ and x₂ each in ten intervals, and the front is the 100 or so patches where both are, which a population must spread over. Then the multimodal g, whose local minima put local fronts parallel to the true one, with the band of feasible g, between 0.5 and 0.5 + ln 2, among them. The four type III spheres, of radius 0.25, are centered at the unit vectors and at (1, 1, 1)/√3, which with this triplet lie between the front and the origin, where the band of g lets no solution be: they don't block the way to it. + +## Representation + +A `Real` genome of 30 genes in [0, 1]. The problem is genoxide's `DasCmop7`, whose fitness is the three objectives and the total constraint violation, 0 when it is feasible; in Python, `gx.problems.DasCmop7()`, with `difficulty=(η, ζ, γ)` or the number of one of the paper's sixteen triplets. Solutions compare by constrained dominance: a feasible solution beats an infeasible one, of two infeasible ones the smaller violation wins, and of two feasible ones Pareto dominance decides. + +## Algorithm + +Two runs, each with a population of 300 for 1,000 generations, 300,000 evaluations (the paper's budget, section 7.1), simulated binary crossover with η = 20 and polynomial mutation at a rate of 1/30 per gene: + +- NSGA-II (Deb, Pratap, Agarwal and Meyarivan, 2002, IEEE Transactions on Evolutionary Computation 6(2): 182-197) with the paper's settings: crossover at a rate of 0.9, mutation with η = 20; +- NSGA-III (Deb and Jain, 2014, IEEE Transactions on Evolutionary Computation 18(4): 577-601) with the 276 reference directions of Das and Dennis's method with 22 divisions, crossover at a rate of 1, and mutation with η = 5, whose steps are larger (a median of about 11% of the range, against 3% with η = 20). + +## Output + +A line per run: the size of its final front, how many of its solutions the problem finds feasible, their IGD+ and hypervolume, and the hypervolume as a share of that of a sample of the optimal front with at least as many points as the population. Then the hypervolumes of the whole front and of the sample. + +The indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 2,000 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest feasible solution of the found front, counting only the objectives in which the solution is worse. Smaller is better; a front of as many points as the population can't cover the 2,000 exactly, and the sample shows what it can. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the volume the feasible solutions dominate up to the reference point (1.1, 1.1, 1.1). Larger is better; the whole front's is computed from 3,000 of its points, and the sample's is what a front of that many points reaches. In Python, `run` evaluates the problem in Rust, so both versions print the same. + +The page plays the runs back side by side, each population as the problem scores it: its feasible non-dominated solutions, its infeasible ones (hollow, at most 100 a frame), and the optimal front, sampled. + +[The project page](https://tachsin.gr/projects/genoxide/examples/das-cmop7) plays these runs back. + +## Good results + +The target: every solution feasible, and 99% of the hypervolume of the sample of 649 points of the front, about what a front of 300 solutions spread like it has. + +With the paper's settings, NSGA-II's run with seed 1 ends with 300 solutions, 300 feasible, IGD+ 0.0190, hypervolume 1.0978, 98.2% of the sample's. NSGA-III's ends with 300 solutions, 300 feasible, IGD+ 0.0136, hypervolume 1.1115, 99.4% of the sample's. Over seeds 1 to 20, NSGA-II ends with IGD+ from 0.0174 to 0.0193 and 98.0% to 98.6% of the sample's hypervolume; with NSGA-III, every run reaches the target, with IGD+ from 0.0124 to 0.0135 and 99.1% to 99.7% of the sample's hypervolume. + +The paper's NSGA-II-CDP ends with a mean IGD of 0.0225 on DAS-CMOP7 with this triplet (its table 5, number 8), and MOEA/D-CDP with 0.102. NSGA-III, whose reference directions spread the population evenly, covers the patches better than NSGA-II's crowding distance does. diff --git a/examples/das_cmop7/main.py b/examples/das_cmop7/main.py new file mode 100644 index 00000000..2aecf4fe --- /dev/null +++ b/examples/das_cmop7/main.py @@ -0,0 +1,95 @@ +"""DAS-CMOP7: minimize three objectives over 30 variables subject to 7 constraints, whose front is +patches of a plane, with NSGA-II and NSGA-III. + +From genoxide's problems.DasCmop7; run evaluates it in Rust. Runs NSGA-II with the paper's settings +and NSGA-III with polynomial mutation with η = 5, a population of 300 for 1,000 generations each, +with the difficulty triplet of the paper's figure 6, (0.5, 0.5, 0.5). Prints each final front's +size, how many of its solutions are feasible, their IGD+ to 2,000 points of the optimal front and +their hypervolume, with the objectives normalized by the front's ideal and nadir points, as a share +of that of a sample of the front with at least as many points as the population; then the +hypervolumes of the whole front and of the sample. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/das_cmop7/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points +REFERENCE = [1.1, 1.1, 1.1] + +problem = gx.problems.DasCmop7() +ideal, nadir = problem.ideal_point, problem.nadir_point +rate = 1 / problem.dimensions + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) +# the sample: at least as many points of the optimal front as the population has, about what a +# front of that many solutions can be +sample = normalized(problem.optimal_front(300)) + + +def run(name, algorithm, fitness, generations): + """Runs ``algorithm`` on ``fitness`` for ``generations``, and prints its final front's size, + how many of its solutions the problem finds feasible, their IGD+ to 2,000 points of the + optimal front and their hypervolume, with normalized objectives, as a share of the + sample's.""" + result = algorithm.run(fitness, generations=generations, on_generation=trace.front(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"{name}, {generations} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(2_000)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + percent = 100 * volume / gx.indicators.hypervolume(sample, REFERENCE) + print( + start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}, " + f"{percent:.1f}% of the sample's" + ) + + +# the paper's settings: NSGA-II, a population of 300 for 1,000 generations +algorithm = gx.Nsga2( + problem.genome, + objectives=problem.objectives, + population_size=300, + crossover=gx.SimulatedBinaryCrossover(20), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-II, η = 20", algorithm, problem, 1_000) +# NSGA-III with 276 reference directions (22 divisions) and polynomial mutation with η = 5 +algorithm = gx.Nsga3( + problem.genome, + objectives=problem.objectives, + reference_directions=gx.das_dennis(3, 22), + population_size=300, + crossover=gx.SimulatedBinaryCrossover(20), + crossover_rate=1.0, + mutation=gx.PolynomialMutation(5, rate=rate), + seed=1, +) +run("NSGA-III, η = 5", algorithm, problem, 1_000) +whole = normalized(problem.optimal_front(3_000)) +print( + f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}; " + f"a sample of {len(sample)} of its points: " + f"{gx.indicators.hypervolume(sample, REFERENCE):.4f}" +) +trace.write() diff --git a/examples/das_cmop7/main.rs b/examples/das_cmop7/main.rs new file mode 100644 index 00000000..c40d0d53 --- /dev/null +++ b/examples/das_cmop7/main.rs @@ -0,0 +1,147 @@ +//! DAS-CMOP7: minimize three objectives over 30 variables subject to 7 constraints, whose front is +//! patches of a plane, with NSGA-II and NSGA-III. +//! +//! From genoxide's `multi::problems::DasCmop7`. Runs NSGA-II with the paper's settings and NSGA-III +//! with polynomial mutation with η = 5, a population of 300 for 1,000 generations each, with the +//! difficulty triplet of the paper's figure 6, (0.5, 0.5, 0.5). Prints each final front's size, how +//! many of its solutions are feasible, their IGD+ to 2,000 points of the optimal front and their +//! hypervolume, with the objectives normalized by the front's ideal and nadir points, as a share of +//! that of a sample of the front with at least as many points as the population; then the +//! hypervolumes of the whole front and of the sample. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example das_cmop7 +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{DasCmop7, MultiProblem}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points +pub const REFERENCE: [f64; 3] = [1.1, 1.1, 1.1]; + +fn main() -> Result<()> { + let problem = DasCmop7::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the sample: at least as many points of the optimal front as the population has, about what + // a front of that many solutions can be + let sample = normalized(&problem, &problem.optimal_front(300).expect("known")); + let rate = 1.0 / problem.variables() as f64; + // the paper's settings: NSGA-II, a population of 300 for 1,000 generations + let algorithm = Nsga2::builder(problem.representation(), [Minimize; 3]) + .population_size(300) + .crossover(SimulatedBinaryCrossover::new(20.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, + "NSGA-II, η = 20", + algorithm, + problem, + 1_000, + &sample, + &mut trace, + )?; + // NSGA-III with 276 reference directions (22 divisions) and polynomial mutation with η = 5 + let algorithm = Nsga3::builder( + problem.representation(), + [Minimize; 3], + multi::das_dennis::<3>(22), + ) + .population_size(300) + .crossover(SimulatedBinaryCrossover::new(20.0)?) + .crossover_rate(1.0) + .mutate(PolynomialMutation::per_gene(rate, 5.0)?) + .seed(1) + .build()?; + run( + &problem, + "NSGA-III, η = 5", + algorithm, + problem, + 1_000, + &sample, + &mut trace, + )?; + let whole = normalized(&problem, &problem.optimal_front(3_000).expect("known")); + println!( + "the whole front: hypervolume {:.4}; a sample of {} of its points: {:.4}", + hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + sample.len(), + hypervolume(&sample, &REFERENCE, &[Minimize; 3]) + ); + trace.write(&problem); + Ok(()) +} + +// runs `algorithm` on `fitness` for `generations`, and prints its final front's size, how many of +// its solutions the problem finds feasible, their IGD+ to 2,000 points of the optimal front and +// their hypervolume, with normalized objectives, as a share of the sample's +fn run( + problem: &DasCmop7, + name: &'static str, + algorithm: A, + fitness: F, + generations: u64, + sample: &[[f64; 3]], + trace: &mut trace::Trace, +) -> Result<()> +where + A: MultiObjectiveAlgorithm<3, Genome = Reals>, + F: MultiFitnessFunction + Sync, +{ + let mut record = trace.front(name, *problem); + let outcome = MultiEngine::new(algorithm, fitness) + .stop_when(Stop::generations(generations)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 3], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "{name}, {generations} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(2_000).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 3]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 3]); + let percent = 100.0 * volume / hypervolume(sample, &REFERENCE, &[Minimize; 3]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the \ + sample's", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &DasCmop7, points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 3]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/das_cmop7/output.txt b/examples/das_cmop7/output.txt new file mode 100644 index 00000000..aea73e55 --- /dev/null +++ b/examples/das_cmop7/output.txt @@ -0,0 +1,3 @@ +NSGA-II, η = 20, 1000 generations: 300 solutions, 300 feasible, IGD+ 0.0190, hypervolume 1.0978, 98.2% of the sample's +NSGA-III, η = 5, 1000 generations: 300 solutions, 300 feasible, IGD+ 0.0136, hypervolume 1.1115, 99.4% of the sample's +the whole front: hypervolume 1.1300; a sample of 649 of its points: 1.1178 diff --git a/examples/das_cmop7/trace.json b/examples/das_cmop7/trace.json new file mode 100644 index 00000000..9cea6711 --- /dev/null +++ b/examples/das_cmop7/trace.json @@ -0,0 +1,26 @@ 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+]} diff --git a/examples/das_cmop7/trace.py b/examples/das_cmop7/trace.py new file mode 100644 index 00000000..25598ebf --- /dev/null +++ b/examples/das_cmop7/trace.py @@ -0,0 +1,215 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: each run's population as the problem scores it, its feasible +non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +front's hypervolume with normalized objectives, in at most 24 generations; a frame's +evaluations are the runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the most infeasible solutions a frame shows +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions and the front's hypervolume + self.series = {} + + def front(self, name): + """The callback for ``run`` that records the run ``name`` after each generation, its + population scored by the problem, which the run may not optimize as it is: None without a + trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + objectives, violations = self.problem.evaluate(progress.population) + front = non_dominated(objectives[violations == 0]) + infeasible = objectives[violations > 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + history.append((progress.evaluations, front.tolist(), infeasible.tolist(), volume)) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(24) + for generation in range(max(map(len, self.series.values()), default=0)): + at = [runs[min(generation, len(runs) - 1)] for runs in self.series.values()] + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at), + "best": None, + "median": None, + "series": {name: record[3] for name, record in zip(self.series, at)}, + "state": { + "panels": [{"front": record[1], "infeasible": record[2]} for record in at] + }, + } + ) + # the front, sampled, for every panel + panel = { + "objectives": ["f1", "f2", "f3"], + "true_front": self.problem.optimal_front(600).tolist(), + } + panels = [{"title": name, "problem": panel} for name in self.series] + whole = self.normalized(self.problem.optimal_front(3_000)) + settings = { + "format": 1, + "example": "das_cmop7", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": list(self.series), + }, + } + write(self.path, settings, frames.to_list()) + + +def non_dominated(points): + """The points that no other point dominates, in their order.""" + keep = [ + not np.any(np.all(points <= point, axis=1) & np.any(points < point, axis=1)) + for point in points + ] + return points[np.array(keep, dtype=bool)] if len(points) else points + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/das_cmop7/trace.rs b/examples/das_cmop7/trace.rs new file mode 100644 index 00000000..17fa83f0 --- /dev/null +++ b/examples/das_cmop7/trace.rs @@ -0,0 +1,307 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: each run's population as the problem scores it, its feasible +//! non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +//! front's hypervolume with normalized objectives, in at most 24 generations; a frame's +//! evaluations are the runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::MultiSnapshot; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{DasCmop7, MultiProblem}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, and the +// hypervolume +type Record = (u64, Vec<[f64; 3]>, Vec<[f64; 3]>, f64); + +// the most infeasible solutions a frame shows +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation: its population scored by + // `problem`, which the run may not optimize as it is + pub fn front( + &mut self, + name: &'static str, + problem: DasCmop7, + ) -> impl FnMut(&MultiSnapshot<'_, Reals, 3>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let population = snapshot.population().as_slice(); + let scores: Vec<([f64; 3], f64)> = population + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = + scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let front = non_dominated(&feasible); + let infeasible = scores.iter().filter(|s| s.1 > 0.0).map(|s| s.0); + let infeasible: Vec<[f64; 3]> = infeasible.take(SHOWN).collect(); + let volume = hypervolume(&normalized(&problem, &front), &REFERENCE, &[Minimize; 3]); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &DasCmop7) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(24); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let (mut panels, mut volumes, mut evaluations) = (Vec::new(), Map::new(), 0); + for (name, history) in &self.series { + let (done, front, infeasible, volume) = &history[generation.min(history.len() - 1)]; + panels.push(json!({ "front": front, "infeasible": infeasible })); + volumes.insert(name.to_string(), json!(volume)); + evaluations += done; + } + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "series": volumes, + "state": { "panels": panels }, + })); + } + // the front, sampled, for every panel + let panel = json!({ + "objectives": ["f1", "f2", "f3"], + "true_front": problem.optimal_front(600).expect("known"), + }); + let panels: Vec = self + .series + .iter() + .map(|(name, _)| json!({ "title": name, "problem": panel })) + .collect(); + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(3_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "das_cmop7", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": names, + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the points that no other point dominates, in their order +fn non_dominated(points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let dominates = + |p: &[f64; 3], q: &[f64; 3]| (0..3).all(|j| p[j] <= q[j]) && (0..3).any(|j| p[j] < q[j]); + let kept = points + .iter() + .filter(|q| !points.iter().any(|p| dominates(p, q))); + kept.copied().collect() +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/das_cmop8/README.md b/examples/das_cmop8/README.md new file mode 100644 index 00000000..61a3294f --- /dev/null +++ b/examples/das_cmop8/README.md @@ -0,0 +1,68 @@ +--- +title: DAS-CMOP8 +category: multi-objective +summary: Minimize three objectives over 30 variables subject to 7 constraints of adjustable difficulty, whose Pareto front is patches of the unit sphere's octant moved to (0.5, with NSGA-II and NSGA-III. +reference: "Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). Difficulty adjustable and scalable constrained multiobjective test problem toolkit. Evolutionary Computation 28(3): 339-378." +reference_url: https://doi.org/10.1162/evco_a_00259 +optimum: "for the difficulty triplet (0.5, 0.5, 0.5), patches of the unit sphere's octant moved to (0.5, 0.5, 0.5); ideal point (0.5, 0.5, 0.5), nadir point (1.5, 1.5, 1.4969); hypervolume 0.7853 (normalized objectives, reference point (1.1, 1.1, 1.1))" +languages: [rust, python] +order: 214 +family: DAS-CMOP +--- + +# DAS-CMOP8 + +## The problem + +Fan et al. (2020) built a toolkit of constrained test problems whose difficulty is set by a triplet (η, ζ, γ) in [0, 1]³: η for diversity (type I constraints, which cut the front into pieces), ζ for feasibility (a type II constraint, a band of distances from the unconstrained front) and γ for convergence (type III constraints, infeasible regions in the objective space). DAS-CMOP7 to DAS-CMOP9 have three objectives; see [DAS-CMOP1](../das_cmop1/) for the two-objective problems. + +DAS-CMOP8 (table 2) has three objectives over 30 variables in [0, 1]: + +```text +f₁ = cos(0.5πx₁) cos(0.5πx₂) + g +f₂ = cos(0.5πx₁) sin(0.5πx₂) + g +f₃ = sin(0.5πx₁) + g +g = 28 + Σⱼ₌₃³⁰ ((xⱼ − 0.5)² − cos(20π(xⱼ − 0.5))) +subject to + sin(20πx₁) − b ≥ 0, cos(20πx₂) − b ≥ 0, b = 2η − 1 (type I) + (e − g)(g − 0.5) ≥ 0, e = 0.5 − ln ζ (type II) + Σⱼ fⱼ² − fₖ² + (fₖ − 1)² − r² ≥ 0, k = 1, 2, 3 (type III) + Σⱼ (fⱼ − 1/√3)² − r² ≥ 0, r = γ/2 (type III) +``` + +All three are minimized. At g = 0 the unconstrained front is the unit sphere's octant. The distance function g is 0 with every xⱼ at 0.5, and has 11²⁸ − 1 local minima, the cosine's, as DTLZ1's. The example uses the triplet with which the paper's figure 6 plots DAS-CMOP8, (0.5, 0.5, 0.5): b = 0, which keeps x₁ in [0, 0.05], [0.1, 0.15], … and x₂ in [0, 0.025], [0.075, 0.125], …; g between 0.5 and 0.5 + ln 2 ≈ 1.193; and spheres of radius 0.25. + +The front depends on the triplet, and the paper samples it; genoxide samples it in the same way, as the first feasible point of each ray α(x₁, x₂) + g (1, 1, 1) from the 41,905 points of Das and Dennis's method with 288 divisions, non-dominated. With this triplet it is patches of the sphere of radius 1 centered at (0.5, 0.5, 0.5), the unit sphere's octant moved out by 0.5 along the diagonal, where x₁ and x₂ are both in the type I intervals. Its ideal point is (0.5, 0.5, 0.5) and its nadir point (1.5, 1.5, 1.4969) (1.4969 for f₃, where x₁ is at most 0.95). + +## What makes it hard + +The grid of patches first: the type I constraints keep x₁ and x₂ each in ten intervals, and the front is the 100 or so patches where both are, which a population must spread over. Then the multimodal g, whose local minima put local fronts parallel to the true one, with the band of feasible g, between 0.5 and 0.5 + ln 2, among them. The four type III spheres, of radius 0.25, are centered at the unit vectors and at (1, 1, 1)/√3, which with this triplet lie between the front and the origin, where the band of g lets no solution be: they don't block the way to it. + +## Representation + +A `Real` genome of 30 genes in [0, 1]. The problem is genoxide's `DasCmop8`, whose fitness is the three objectives and the total constraint violation, 0 when it is feasible; in Python, `gx.problems.DasCmop8()`, with `difficulty=(η, ζ, γ)` or the number of one of the paper's sixteen triplets. Solutions compare by constrained dominance: a feasible solution beats an infeasible one, of two infeasible ones the smaller violation wins, and of two feasible ones Pareto dominance decides. + +## Algorithm + +Two runs, each with a population of 300 for 1,000 generations, 300,000 evaluations (the paper's budget, section 7.1), simulated binary crossover with η = 20 and polynomial mutation at a rate of 1/30 per gene: + +- NSGA-II (Deb, Pratap, Agarwal and Meyarivan, 2002, IEEE Transactions on Evolutionary Computation 6(2): 182-197) with the paper's settings: crossover at a rate of 0.9, mutation with η = 20; +- NSGA-III (Deb and Jain, 2014, IEEE Transactions on Evolutionary Computation 18(4): 577-601) with the 276 reference directions of Das and Dennis's method with 22 divisions, crossover at a rate of 1, and mutation with η = 5, whose steps are larger (a median of about 11% of the range, against 3% with η = 20). + +## Output + +A line per run: the size of its final front, how many of its solutions the problem finds feasible, their IGD+ and hypervolume, and the hypervolume as a share of that of a sample of the optimal front with at least as many points as the population. Then the hypervolumes of the whole front and of the sample. + +The indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 2,000 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest feasible solution of the found front, counting only the objectives in which the solution is worse. Smaller is better; a front of as many points as the population can't cover the 2,000 exactly, and the sample shows what it can. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the volume the feasible solutions dominate up to the reference point (1.1, 1.1, 1.1). Larger is better; the whole front's is computed from 3,000 of its points, and the sample's is what a front of that many points reaches. In Python, `run` evaluates the problem in Rust, so both versions print the same. + +The page plays the runs back side by side, each population as the problem scores it: its feasible non-dominated solutions, its infeasible ones (hollow, at most 100 a frame), and the optimal front, sampled. + +[The project page](https://tachsin.gr/projects/genoxide/examples/das-cmop8) plays these runs back. + +## Good results + +The target: every solution feasible, and 99% of the hypervolume of the sample of 346 points of the front, about what a front of 300 solutions spread like it has. + +With the paper's settings, NSGA-II's run with seed 1 ends with 300 solutions, 300 feasible, IGD+ 0.0158, hypervolume 0.7432, 97.2% of the sample's. NSGA-III's ends with 300 solutions, 300 feasible, IGD+ 0.0111, hypervolume 0.7603, 99.4% of the sample's. Over seeds 1 to 20, NSGA-II ends with IGD+ from 0.0147 to 0.0172 and 96.8% to 97.8% of the sample's hypervolume; with NSGA-III, every run reaches the target, with IGD+ from 0.0099 to 0.0113 and 99.2% to 99.9% of the sample's hypervolume. + +The paper's NSGA-II-CDP ends with a mean IGD of 0.0286 on DAS-CMOP8 with this triplet (its table 5, number 8), and MOEA/D-CDP with 0.0618. NSGA-III, whose reference directions spread the population evenly, covers the patches better than NSGA-II's crowding distance does. diff --git a/examples/das_cmop8/main.py b/examples/das_cmop8/main.py new file mode 100644 index 00000000..faf61e38 --- /dev/null +++ b/examples/das_cmop8/main.py @@ -0,0 +1,95 @@ +"""DAS-CMOP8: minimize three objectives over 30 variables subject to 7 constraints, whose front is +patches of a sphere, with NSGA-II and NSGA-III. + +From genoxide's problems.DasCmop8; run evaluates it in Rust. Runs NSGA-II with the paper's settings +and NSGA-III with polynomial mutation with η = 5, a population of 300 for 1,000 generations each, +with the difficulty triplet of the paper's figure 6, (0.5, 0.5, 0.5). Prints each final front's +size, how many of its solutions are feasible, their IGD+ to 2,000 points of the optimal front and +their hypervolume, with the objectives normalized by the front's ideal and nadir points, as a share +of that of a sample of the front with at least as many points as the population; then the +hypervolumes of the whole front and of the sample. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/das_cmop8/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points +REFERENCE = [1.1, 1.1, 1.1] + +problem = gx.problems.DasCmop8() +ideal, nadir = problem.ideal_point, problem.nadir_point +rate = 1 / problem.dimensions + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) +# the sample: at least as many points of the optimal front as the population has, about what a +# front of that many solutions can be +sample = normalized(problem.optimal_front(300)) + + +def run(name, algorithm, fitness, generations): + """Runs ``algorithm`` on ``fitness`` for ``generations``, and prints its final front's size, + how many of its solutions the problem finds feasible, their IGD+ to 2,000 points of the + optimal front and their hypervolume, with normalized objectives, as a share of the + sample's.""" + result = algorithm.run(fitness, generations=generations, on_generation=trace.front(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"{name}, {generations} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(2_000)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + percent = 100 * volume / gx.indicators.hypervolume(sample, REFERENCE) + print( + start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}, " + f"{percent:.1f}% of the sample's" + ) + + +# the paper's settings: NSGA-II, a population of 300 for 1,000 generations +algorithm = gx.Nsga2( + problem.genome, + objectives=problem.objectives, + population_size=300, + crossover=gx.SimulatedBinaryCrossover(20), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-II, η = 20", algorithm, problem, 1_000) +# NSGA-III with 276 reference directions (22 divisions) and polynomial mutation with η = 5 +algorithm = gx.Nsga3( + problem.genome, + objectives=problem.objectives, + reference_directions=gx.das_dennis(3, 22), + population_size=300, + crossover=gx.SimulatedBinaryCrossover(20), + crossover_rate=1.0, + mutation=gx.PolynomialMutation(5, rate=rate), + seed=1, +) +run("NSGA-III, η = 5", algorithm, problem, 1_000) +whole = normalized(problem.optimal_front(3_000)) +print( + f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}; " + f"a sample of {len(sample)} of its points: " + f"{gx.indicators.hypervolume(sample, REFERENCE):.4f}" +) +trace.write() diff --git a/examples/das_cmop8/main.rs b/examples/das_cmop8/main.rs new file mode 100644 index 00000000..f624c508 --- /dev/null +++ b/examples/das_cmop8/main.rs @@ -0,0 +1,147 @@ +//! DAS-CMOP8: minimize three objectives over 30 variables subject to 7 constraints, whose front is +//! patches of a sphere, with NSGA-II and NSGA-III. +//! +//! From genoxide's `multi::problems::DasCmop8`. Runs NSGA-II with the paper's settings and NSGA-III +//! with polynomial mutation with η = 5, a population of 300 for 1,000 generations each, with the +//! difficulty triplet of the paper's figure 6, (0.5, 0.5, 0.5). Prints each final front's size, how +//! many of its solutions are feasible, their IGD+ to 2,000 points of the optimal front and their +//! hypervolume, with the objectives normalized by the front's ideal and nadir points, as a share of +//! that of a sample of the front with at least as many points as the population; then the +//! hypervolumes of the whole front and of the sample. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example das_cmop8 +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{DasCmop8, MultiProblem}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points +pub const REFERENCE: [f64; 3] = [1.1, 1.1, 1.1]; + +fn main() -> Result<()> { + let problem = DasCmop8::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the sample: at least as many points of the optimal front as the population has, about what + // a front of that many solutions can be + let sample = normalized(&problem, &problem.optimal_front(300).expect("known")); + let rate = 1.0 / problem.variables() as f64; + // the paper's settings: NSGA-II, a population of 300 for 1,000 generations + let algorithm = Nsga2::builder(problem.representation(), [Minimize; 3]) + .population_size(300) + .crossover(SimulatedBinaryCrossover::new(20.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, + "NSGA-II, η = 20", + algorithm, + problem, + 1_000, + &sample, + &mut trace, + )?; + // NSGA-III with 276 reference directions (22 divisions) and polynomial mutation with η = 5 + let algorithm = Nsga3::builder( + problem.representation(), + [Minimize; 3], + multi::das_dennis::<3>(22), + ) + .population_size(300) + .crossover(SimulatedBinaryCrossover::new(20.0)?) + .crossover_rate(1.0) + .mutate(PolynomialMutation::per_gene(rate, 5.0)?) + .seed(1) + .build()?; + run( + &problem, + "NSGA-III, η = 5", + algorithm, + problem, + 1_000, + &sample, + &mut trace, + )?; + let whole = normalized(&problem, &problem.optimal_front(3_000).expect("known")); + println!( + "the whole front: hypervolume {:.4}; a sample of {} of its points: {:.4}", + hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + sample.len(), + hypervolume(&sample, &REFERENCE, &[Minimize; 3]) + ); + trace.write(&problem); + Ok(()) +} + +// runs `algorithm` on `fitness` for `generations`, and prints its final front's size, how many of +// its solutions the problem finds feasible, their IGD+ to 2,000 points of the optimal front and +// their hypervolume, with normalized objectives, as a share of the sample's +fn run( + problem: &DasCmop8, + name: &'static str, + algorithm: A, + fitness: F, + generations: u64, + sample: &[[f64; 3]], + trace: &mut trace::Trace, +) -> Result<()> +where + A: MultiObjectiveAlgorithm<3, Genome = Reals>, + F: MultiFitnessFunction + Sync, +{ + let mut record = trace.front(name, *problem); + let outcome = MultiEngine::new(algorithm, fitness) + .stop_when(Stop::generations(generations)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 3], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "{name}, {generations} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(2_000).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 3]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 3]); + let percent = 100.0 * volume / hypervolume(sample, &REFERENCE, &[Minimize; 3]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the \ + sample's", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &DasCmop8, points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 3]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/das_cmop8/output.txt b/examples/das_cmop8/output.txt new file mode 100644 index 00000000..d55b892b --- /dev/null +++ b/examples/das_cmop8/output.txt @@ -0,0 +1,3 @@ +NSGA-II, η = 20, 1000 generations: 300 solutions, 300 feasible, IGD+ 0.0158, hypervolume 0.7432, 97.2% of the sample's +NSGA-III, η = 5, 1000 generations: 300 solutions, 300 feasible, IGD+ 0.0111, hypervolume 0.7603, 99.4% of the sample's +the whole front: hypervolume 0.7853; a sample of 346 of its points: 0.7647 diff --git a/examples/das_cmop8/trace.json b/examples/das_cmop8/trace.json new file mode 100644 index 00000000..f12513b5 --- /dev/null +++ b/examples/das_cmop8/trace.json @@ -0,0 +1,26 @@ 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+]} diff --git a/examples/das_cmop8/trace.py b/examples/das_cmop8/trace.py new file mode 100644 index 00000000..f79804bc --- /dev/null +++ b/examples/das_cmop8/trace.py @@ -0,0 +1,215 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: each run's population as the problem scores it, its feasible +non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +front's hypervolume with normalized objectives, in at most 24 generations; a frame's +evaluations are the runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the most infeasible solutions a frame shows +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions and the front's hypervolume + self.series = {} + + def front(self, name): + """The callback for ``run`` that records the run ``name`` after each generation, its + population scored by the problem, which the run may not optimize as it is: None without a + trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + objectives, violations = self.problem.evaluate(progress.population) + front = non_dominated(objectives[violations == 0]) + infeasible = objectives[violations > 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + history.append((progress.evaluations, front.tolist(), infeasible.tolist(), volume)) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(24) + for generation in range(max(map(len, self.series.values()), default=0)): + at = [runs[min(generation, len(runs) - 1)] for runs in self.series.values()] + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at), + "best": None, + "median": None, + "series": {name: record[3] for name, record in zip(self.series, at)}, + "state": { + "panels": [{"front": record[1], "infeasible": record[2]} for record in at] + }, + } + ) + # the front, sampled, for every panel + panel = { + "objectives": ["f1", "f2", "f3"], + "true_front": self.problem.optimal_front(600).tolist(), + } + panels = [{"title": name, "problem": panel} for name in self.series] + whole = self.normalized(self.problem.optimal_front(3_000)) + settings = { + "format": 1, + "example": "das_cmop8", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": list(self.series), + }, + } + write(self.path, settings, frames.to_list()) + + +def non_dominated(points): + """The points that no other point dominates, in their order.""" + keep = [ + not np.any(np.all(points <= point, axis=1) & np.any(points < point, axis=1)) + for point in points + ] + return points[np.array(keep, dtype=bool)] if len(points) else points + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/das_cmop8/trace.rs b/examples/das_cmop8/trace.rs new file mode 100644 index 00000000..88dd598a --- /dev/null +++ b/examples/das_cmop8/trace.rs @@ -0,0 +1,307 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: each run's population as the problem scores it, its feasible +//! non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +//! front's hypervolume with normalized objectives, in at most 24 generations; a frame's +//! evaluations are the runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::MultiSnapshot; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{DasCmop8, MultiProblem}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, and the +// hypervolume +type Record = (u64, Vec<[f64; 3]>, Vec<[f64; 3]>, f64); + +// the most infeasible solutions a frame shows +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation: its population scored by + // `problem`, which the run may not optimize as it is + pub fn front( + &mut self, + name: &'static str, + problem: DasCmop8, + ) -> impl FnMut(&MultiSnapshot<'_, Reals, 3>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let population = snapshot.population().as_slice(); + let scores: Vec<([f64; 3], f64)> = population + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = + scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let front = non_dominated(&feasible); + let infeasible = scores.iter().filter(|s| s.1 > 0.0).map(|s| s.0); + let infeasible: Vec<[f64; 3]> = infeasible.take(SHOWN).collect(); + let volume = hypervolume(&normalized(&problem, &front), &REFERENCE, &[Minimize; 3]); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &DasCmop8) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(24); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let (mut panels, mut volumes, mut evaluations) = (Vec::new(), Map::new(), 0); + for (name, history) in &self.series { + let (done, front, infeasible, volume) = &history[generation.min(history.len() - 1)]; + panels.push(json!({ "front": front, "infeasible": infeasible })); + volumes.insert(name.to_string(), json!(volume)); + evaluations += done; + } + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "series": volumes, + "state": { "panels": panels }, + })); + } + // the front, sampled, for every panel + let panel = json!({ + "objectives": ["f1", "f2", "f3"], + "true_front": problem.optimal_front(600).expect("known"), + }); + let panels: Vec = self + .series + .iter() + .map(|(name, _)| json!({ "title": name, "problem": panel })) + .collect(); + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(3_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "das_cmop8", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": names, + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the points that no other point dominates, in their order +fn non_dominated(points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let dominates = + |p: &[f64; 3], q: &[f64; 3]| (0..3).all(|j| p[j] <= q[j]) && (0..3).any(|j| p[j] < q[j]); + let kept = points + .iter() + .filter(|q| !points.iter().any(|p| dominates(p, q))); + kept.copied().collect() +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/das_cmop9/README.md b/examples/das_cmop9/README.md new file mode 100644 index 00000000..1815a117 --- /dev/null +++ b/examples/das_cmop9/README.md @@ -0,0 +1,72 @@ +--- +title: DAS-CMOP9 +category: multi-objective +summary: Minimize three objectives over 30 variables subject to 7 constraints of adjustable difficulty, whose Pareto front is patches of a sphere, with linked variables, with MOEA/D-DE, which reaches the front and spreads on it as well as its 300 subproblems allow, and NSGA-II, which covers only part of it. +reference: "Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). Difficulty adjustable and scalable constrained multiobjective test problem toolkit. Evolutionary Computation 28(3): 339-378." +reference_url: https://doi.org/10.1162/evco_a_00259 +optimum: "for the difficulty triplet (0.5, 0.5, 0.5), DAS-CMOP8's, patches of the unit sphere's octant moved to (0.5, 0.5, 0.5); ideal point (0.5, 0.5, 0.5), nadir point (1.5, 1.5, 1.4969); hypervolume 0.7853 (normalized objectives, reference point (1.1, 1.1, 1.1))" +languages: [rust, python] +order: 215 +family: DAS-CMOP +--- + +# DAS-CMOP9 + +## The problem + +Fan et al. (2020) built a toolkit of constrained test problems whose difficulty is set by a triplet (η, ζ, γ) in [0, 1]³: η for diversity (type I constraints, which cut the front into pieces), ζ for feasibility (a type II constraint, a band of distances from the unconstrained front) and γ for convergence (type III constraints, infeasible regions in the objective space). DAS-CMOP7 to DAS-CMOP9 have three objectives; see [DAS-CMOP1](../das_cmop1/) for the two-objective problems. + +DAS-CMOP9 (table 2) has three objectives over 30 variables in [0, 1]: + +```text +f₁ = cos(0.5πx₁) cos(0.5πx₂) + g +f₂ = cos(0.5πx₁) sin(0.5πx₂) + g +f₃ = sin(0.5πx₁) + g +g = Σⱼ₌₃³⁰ (xⱼ − cos(0.25jπ(x₁ + x₂)/30))² +subject to + sin(20πx₁) − b ≥ 0, cos(20πx₂) − b ≥ 0, b = 2η − 1 (type I) + (e − g)(g − 0.5) ≥ 0, e = 0.5 − ln ζ (type II) + Σⱼ fⱼ² − fₖ² + (fₖ − 1)² − r² ≥ 0, k = 1, 2, 3 (type III) + Σⱼ (fⱼ − 1/√3)² − r² ≥ 0, r = γ/2 (type III) +``` + +All three are minimized. At g = 0 the unconstrained front is the unit sphere's octant. The distance function g is 0 where each xⱼ is its own cosine of x₁ + x₂, different for each j: the variables are linked to the position variables. The example uses the triplet with which the paper's figure 6 plots DAS-CMOP9, (0.5, 0.5, 0.5): b = 0, which keeps x₁ in [0, 0.05], [0.1, 0.15], … and x₂ in [0, 0.025], [0.075, 0.125], …; g between 0.5 and 0.5 + ln 2 ≈ 1.193; and spheres of radius 0.25. + +The front depends on the triplet, and the paper samples it; genoxide samples it in the same way, as the first feasible point of each ray α(x₁, x₂) + g (1, 1, 1) from the 41,905 points of Das and Dennis's method with 288 divisions, non-dominated. With this triplet it is DAS-CMOP8's: patches of the sphere of radius 1 centered at (0.5, 0.5, 0.5) where x₁ and x₂ are both in the type I intervals. Its ideal point is (0.5, 0.5, 0.5) and its nadir point (1.5, 1.5, 1.4969) (1.4969 for f₃, where x₁ is at most 0.95). + +## What makes it hard + +The linked variables. A solution on the front at (x₁, x₂) has every other variable at its own cosine of x₁ + x₂: to move along the front, all 28 must move together, each by its own amount. Simulated binary crossover and polynomial mutation change each variable on its own, so a population that has settled on some values of x₁ + x₂ keeps them, and covers only part of the front's patches. Differential evolution moves them together, as its steps are differences between solutions of the population: the paper's MOEA/D-CDP uses it, and so does MOEA/D-DE here. Then, as for DAS-CMOP8, the band of feasible g and the grid of patches. + +## Representation + +A `Real` genome of 30 genes in [0, 1]. The problem is genoxide's `DasCmop9`, whose fitness is the three objectives and the total constraint violation, 0 when it is feasible; in Python, `gx.problems.DasCmop9()`, with `difficulty=(η, ζ, γ)` or the number of one of the paper's sixteen triplets. Solutions compare by constrained dominance: a feasible solution beats an infeasible one, of two infeasible ones the smaller violation wins, and of two feasible ones Pareto dominance decides. + +## Algorithm + +Two runs, each with the paper's population of 300 for 1,000 generations, 300,000 evaluations (section 7.1), and polynomial mutation with η = 20 at a rate of 1/30 per gene: + +- MOEA/D with differential evolution, MOEA/D-DE (Li and Zhang, 2009, IEEE Transactions on Evolutionary Computation 13(2): 284-302): a subproblem for each of the 300 weight vectors of Das and Dennis's method with 23 divisions, each the weighted Tchebycheff distance to the ideal point, with the paper's 30 neighbours (0.1 N) and at most 2 replacements per child; a child is its subproblem's solution moved by F = 0.5 times the difference of two parents, every gene (CR = 1), the parents from the neighbourhood with probability δ = 0.2, where the paper and Li and Zhang use 0.9 (DAS-CMOP1's page says why); +- NSGA-II (Deb, Pratap, Agarwal and Meyarivan, 2002, IEEE Transactions on Evolutionary Computation 6(2): 182-197) with the paper's settings, simulated binary crossover with η = 20 at a rate of 0.9, as the contrast. + +Both compare solutions by constrained dominance. + +## Output + +A line per run: the size of its final front, how many of its solutions the problem finds feasible, their IGD+ and hypervolume, the hypervolume as a share of that of a sample of the optimal front with at least as many points as the population, and the median and largest distance from its solutions to the nearest point of a dense sample of the front (44,643 points). Then a line for what MOEA/D's decomposition can reach: for each of its 300 weight vectors, the point of the dense sample with the least Tchebycheff value, the best that subproblem can have, and the same indicators for these points; and the same with 666 weight vectors (35 divisions). Then the hypervolumes of the whole front and of the sample. + +The indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 2,000 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest feasible solution of the found front, counting only the objectives in which the solution is worse. Smaller is better; a front of as many points as the population can't cover the 2,000 exactly, and the sample shows what it can. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the volume the feasible solutions dominate up to the reference point (1.1, 1.1, 1.1). Larger is better; the whole front's is computed from 3,000 of its points, and the sample's is what a front of that many points reaches. In Python, `run` evaluates the problem in Rust, so both versions print the same. + +The page plays the runs back side by side, each population as the problem scores it: its feasible non-dominated solutions, its infeasible ones (hollow, at most 100 a frame), and the optimal front, sampled. + +[The project page](https://tachsin.gr/projects/genoxide/examples/das-cmop9) plays these runs back. + +## Good results + +The target: every solution feasible, and 99% of the hypervolume of the sample of 346 points of the front, about what a front of 300 solutions spread like it has. + +MOEA/D-DE's run with seed 1 ends with 196 solutions, 196 feasible, IGD+ 0.0158, hypervolume 0.7484, 97.9% of the sample's; half of its solutions are within 0.0018 of the front, and the farthest 0.0200 away. NSGA-II's ends with 300 solutions, 300 feasible, IGD+ 0.2006, hypervolume 0.4880, 63.8% of the sample's: as close to the front (a median of 0.0017), but on only part of it. Over seeds 1 to 20, MOEA/D-DE ends with IGD+ from 0.0151 to 0.0167 and 97.3% to 98.0% of the sample's hypervolume; half of each run's solutions within 0.0018 to 0.0023 of the front, 89% to 97% of them within 0.01, and the farthest 0.014 to 0.047 away. NSGA-II ends with IGD+ from 0.0158 to 0.3408 and 45% to 97% of the sample's hypervolume. With δ = 0.9, MOEA/D-DE ends with IGD+ from 0.0152 to 0.5144 and 20% to 98%. + +MOEA/D-DE reaches the front, and spreads on it as well as its decomposition allows, but no run reaches the 99% target, because the target doesn't fit 300 Tchebycheff subproblems. Each subproblem can at best hold the front's point with its least Tchebycheff value. Those best points, one per weight vector, are only 223 distinct points, as several weight vectors share one on the patched front. Together they have IGD+ 0.0184 and 96.7% of the sample's hypervolume: every run of MOEA/D-DE does better on both. With 666 weight vectors, the best points reach 99.4%: the target asks for a front spread evenly by length, as the sample is, which the 300 subproblems can't give. The few solutions farther from the front than 0.01 haven't converged yet: with seed 1, the farthest is dominated by 81 points of the dense sample. + +The paper's NSGA-II-CDP ends with a mean IGD of 0.227 on DAS-CMOP9 with this triplet (its table 5, number 8), and MOEA/D-CDP, with differential evolution, with 0.0767. diff --git a/examples/das_cmop9/main.py b/examples/das_cmop9/main.py new file mode 100644 index 00000000..994f77e6 --- /dev/null +++ b/examples/das_cmop9/main.py @@ -0,0 +1,148 @@ +"""DAS-CMOP9: minimize three objectives over 30 variables subject to 7 constraints, whose front is +patches of a sphere, with linked variables, with MOEA/D-DE and NSGA-II. + +From genoxide's problems.DasCmop9; run evaluates it in Rust. Runs MOEA/D with differential +evolution (MOEA/D-DE) and, as a contrast, NSGA-II with the paper's settings, a population of 300 +for 1,000 generations each, with the difficulty triplet of the paper's figure 6, (0.5, 0.5, 0.5). +Prints each final front's size, how many of its solutions are feasible, their IGD+ to 2,000 points +of the optimal front and their hypervolume, with the objectives normalized by the front's ideal and +nadir points, as a share of that of a sample of the front with at least as many points as the +population, and the median and largest distance from them to a dense sample of the front. Then the +same indicators for the best point of the dense sample for each of MOEA/D's 300 weight vectors, and +for 666, and the hypervolumes of the whole front and of the sample. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/das_cmop9/main.py +""" + +import numpy as np + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points +REFERENCE = [1.1, 1.1, 1.1] + +# the paper's population and evaluations: 300 for 1,000 generations; MOEA/D's 300 weight vectors +# are Das and Dennis's with 23 divisions +POPULATION = 300 +GENERATIONS = 1_000 +DIVISIONS = 23 + +problem = gx.problems.DasCmop9() +ideal, nadir = problem.ideal_point, problem.nadir_point +rate = 1 / problem.dimensions + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) +# the sample: at least as many points of the optimal front as the population has, about what a +# front of that many solutions can be +sample = normalized(problem.optimal_front(POPULATION)) +# a dense sample of the front, at least 40,000 points +dense = problem.optimal_front(40_000) + + +def indicators(points): + """The IGD+ of ``points`` to 2,000 points of the optimal front, their hypervolume, and that as a + percentage of the sample's, with normalized objectives.""" + found = normalized(points) + distance = gx.indicators.igd_plus(found, normalized(problem.optimal_front(2_000))) + volume = gx.indicators.hypervolume(found, REFERENCE) + return distance, volume, 100 * volume / gx.indicators.hypervolume(sample, REFERENCE) + + +def distances(points): + """The median and the largest distance from a point of ``points`` to the nearest point of the + dense sample, with normalized objectives.""" + near = normalized(dense) + found = [np.sqrt(((near - p) * (near - p)).sum(axis=1).min()) for p in normalized(points)] + return np.median(found), max(found) + + +def per_weight(weights): + """For each weight vector, the point of the dense sample that minimizes MOEA/D's Tchebycheff + value maxⱼ wⱼ |fⱼ − zⱼ| around the ideal point z (of two equal, the one with the least + Σ (fⱼ − zⱼ), then the first), each point once, sorted.""" + shifted = dense - ideal + sums = (shifted[:, 0] + shifted[:, 1]) + shifted[:, 2] + best = [] + for w in weights: + values = (w * np.abs(shifted)).max(axis=1) + tied = np.flatnonzero(values == values.min()) + best.append(dense[tied[np.argmin(sums[tied])]]) + return np.unique(np.array(best), axis=0) + + +def run(name, algorithm): + """Runs ``algorithm`` for the paper's 1,000 generations, and prints its final front's size, how + many of its solutions the problem finds feasible, their IGD+ to 2,000 points of the optimal + front and their hypervolume, with normalized objectives, as a share of the sample's, and the + median and largest distance from them to the dense sample of the front.""" + result = algorithm.run(problem, generations=GENERATIONS, on_generation=trace.front(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"{name}, {GENERATIONS} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + distance, volume, percent = indicators(feasible) + median, largest = distances(feasible) + print( + start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}, " + f"{percent:.1f}% of the sample's; from the front: {median:.4f} (median), " + f"{largest:.4f} (farthest)" + ) + + +# MOEA/D-DE: a subproblem per weight vector, 30 neighbours (the paper's 0.1 N), parents from the +# neighbourhood with probability 0.2, differential evolution with F = 0.5 and CR = 1 +moead = gx.Moead( + problem.genome, + objectives=problem.objectives, + weights=gx.das_dennis(3, DIVISIONS), + neighbors=30, + neighbor_mating=0.2, + crossover=gx.DifferentialEvolutionCrossover(f=0.5, cr=1.0), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("MOEA/D-DE", moead) +# the contrast: NSGA-II with the paper's settings +nsga2 = gx.Nsga2( + problem.genome, + objectives=problem.objectives, + population_size=POPULATION, + crossover=gx.SimulatedBinaryCrossover(20), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-II", nsga2) +# what a decomposition into Tchebycheff subproblems can reach: each weight vector's best point of +# the dense sample, with MOEA/D's 300 vectors and with 666 (35 divisions) +for divisions in (DIVISIONS, 35): + weights = gx.das_dennis(3, divisions) + best = per_weight(weights) + distance, volume, percent = indicators(best) + print( + f"the best point of the front for each of {len(weights)} weight vectors: {len(best)} " + f"points, IGD+ {distance:.4f}, hypervolume {volume:.4f}, {percent:.1f}% of the sample's" + ) +whole = normalized(problem.optimal_front(3_000)) +print( + f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}; " + f"a sample of {len(sample)} of its points: " + f"{gx.indicators.hypervolume(sample, REFERENCE):.4f}" +) +trace.write() diff --git a/examples/das_cmop9/main.rs b/examples/das_cmop9/main.rs new file mode 100644 index 00000000..b44c49f4 --- /dev/null +++ b/examples/das_cmop9/main.rs @@ -0,0 +1,219 @@ +//! DAS-CMOP9: minimize three objectives over 30 variables subject to 7 constraints, whose front is +//! patches of a sphere, with linked variables, with MOEA/D-DE and NSGA-II. +//! +//! From genoxide's `multi::problems::DasCmop9`. Runs MOEA/D with differential evolution +//! (MOEA/D-DE) and, as a contrast, NSGA-II with the paper's settings, a population of 300 for +//! 1,000 generations each, with the difficulty triplet of the paper's figure 6, (0.5, 0.5, 0.5). +//! Prints each final front's size, how many of its solutions are feasible, their IGD+ to 2,000 +//! points of the optimal front and their hypervolume, with the objectives normalized by the +//! front's ideal and nadir points, as a share of that of a sample of the front with at least as +//! many points as the population, and the median and largest distance from them to a dense sample +//! of the front. Then the same indicators for the best point of the dense sample for each of +//! MOEA/D's 300 weight vectors, and for 666, and the hypervolumes of the whole front and of the +//! sample. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example das_cmop9 +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{DasCmop9, MultiProblem}; +use genoxide::multi::{DifferentialEvolutionCrossover, MultiFitnessFunction}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points +pub const REFERENCE: [f64; 3] = [1.1, 1.1, 1.1]; + +// the paper's population and evaluations: 300 for 1,000 generations; MOEA/D's 300 weight vectors +// are Das and Dennis's with 23 divisions +const POPULATION: usize = 300; +const GENERATIONS: u64 = 1_000; +const DIVISIONS: usize = 23; + +fn main() -> Result<()> { + let problem = DasCmop9::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the sample: at least as many points of the optimal front as the population has, about what + // a front of that many solutions can be + let sample = normalized(&problem, &problem.optimal_front(POPULATION).expect("known")); + // a dense sample of the front, at least 40,000 points + let dense = problem.optimal_front(40_000).expect("known"); + let rate = 1.0 / problem.variables() as f64; + // MOEA/D-DE: a subproblem per weight vector, 30 neighbours (the paper's 0.1 N), parents from + // the neighbourhood with probability 0.2, differential evolution with F = 0.5 and CR = 1 + let moead = Moead::builder( + problem.representation(), + [Minimize; 3], + multi::das_dennis::<3>(DIVISIONS), + ) + .neighbors(30) + .neighbor_mating(0.2) + .crossover(DifferentialEvolutionCrossover::new(0.5, 1.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run(&problem, "MOEA/D-DE", moead, &sample, &dense, &mut trace)?; + // the contrast: NSGA-II with the paper's settings + let nsga2 = Nsga2::builder(problem.representation(), [Minimize; 3]) + .population_size(POPULATION) + .crossover(SimulatedBinaryCrossover::new(20.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run(&problem, "NSGA-II", nsga2, &sample, &dense, &mut trace)?; + // what a decomposition into Tchebycheff subproblems can reach: each weight vector's best point + // of the dense sample, with MOEA/D's 300 vectors and with 666 (35 divisions) + for divisions in [DIVISIONS, 35] { + let weights = multi::das_dennis::<3>(divisions); + let best = per_weight(&problem, &dense, &weights); + let (distance, volume, percent) = indicators(&problem, &best, &sample); + println!( + "the best point of the front for each of {} weight vectors: {} points, IGD+ \ + {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the sample's", + weights.len(), + best.len() + ); + } + let whole = normalized(&problem, &problem.optimal_front(3_000).expect("known")); + println!( + "the whole front: hypervolume {:.4}; a sample of {} of its points: {:.4}", + hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + sample.len(), + hypervolume(&sample, &REFERENCE, &[Minimize; 3]) + ); + trace.write(&problem); + Ok(()) +} + +// runs `algorithm` for the paper's 1,000 generations, and prints its final front's size, how many +// of its solutions the problem finds feasible, their IGD+ to 2,000 points of the optimal front and +// their hypervolume, with normalized objectives, as a share of the sample's, and the median and +// largest distance from them to the dense sample of the front +fn run( + problem: &DasCmop9, + name: &'static str, + algorithm: A, + sample: &[[f64; 3]], + dense: &[[f64; 3]], + trace: &mut trace::Trace, +) -> Result<()> +where + A: MultiObjectiveAlgorithm<3, Genome = Reals>, +{ + let mut record = trace.front(name, *problem); + let outcome = MultiEngine::new(algorithm, *problem) + .stop_when(Stop::generations(GENERATIONS)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 3], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "{name}, {GENERATIONS} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let (distance, volume, percent) = indicators(problem, &feasible, sample); + let (median, farthest) = distances(problem, &feasible, dense); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the \ + sample's; from the front: {median:.4} (median), {farthest:.4} (farthest)", + feasible.len() + ); + Ok(()) +} + +// the IGD+ of `points` to 2,000 points of the optimal front, their hypervolume, and that as a +// percentage of the sample's, with normalized objectives +fn indicators(problem: &DasCmop9, points: &[[f64; 3]], sample: &[[f64; 3]]) -> (f64, f64, f64) { + let found = normalized(problem, points); + let optimal = normalized(problem, &problem.optimal_front(2_000).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 3]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 3]); + let percent = 100.0 * volume / hypervolume(sample, &REFERENCE, &[Minimize; 3]); + (distance, volume, percent) +} + +// the median and the largest distance from a point of `points` to the nearest point of `dense`, +// with normalized objectives +fn distances(problem: &DasCmop9, points: &[[f64; 3]], dense: &[[f64; 3]]) -> (f64, f64) { + let dense = normalized(problem, dense); + let squared = + |p: &[f64; 3], q: &[f64; 3]| -> f64 { (0..3).map(|j| (p[j] - q[j]) * (p[j] - q[j])).sum() }; + let mut distances: Vec = normalized(problem, points) + .iter() + .map(|p| { + dense + .iter() + .map(|q| squared(p, q)) + .fold(f64::INFINITY, f64::min) + .sqrt() + }) + .collect(); + distances.sort_by(f64::total_cmp); + let n = distances.len(); + let median = if n % 2 == 1 { + distances[n / 2] + } else { + (distances[n / 2 - 1] + distances[n / 2]) / 2.0 + }; + (median, distances[n - 1]) +} + +// for each weight vector, the point of `dense` that minimizes MOEA/D's Tchebycheff value +// maxⱼ wⱼ |fⱼ − zⱼ| around the ideal point z (of two equal, the one with the least Σ (fⱼ − zⱼ), +// then the first), each point once +fn per_weight(problem: &DasCmop9, dense: &[[f64; 3]], weights: &[[f64; 3]]) -> Vec<[f64; 3]> { + let ideal = problem.ideal_point().expect("known"); + let mut best: Vec<[f64; 3]> = weights + .iter() + .map(|w| { + let mut chosen = (f64::INFINITY, f64::INFINITY, 0); + for (i, f) in dense.iter().enumerate() { + let value = (0..3) + .map(|j| w[j] * (f[j] - ideal[j]).abs()) + .fold(f64::NEG_INFINITY, f64::max); + let sum = (f[0] - ideal[0]) + (f[1] - ideal[1]) + (f[2] - ideal[2]); + if value < chosen.0 || (value == chosen.0 && sum < chosen.1) { + chosen = (value, sum, i); + } + } + dense[chosen.2] + }) + .collect(); + best.sort_by(|a, b| { + a[0].total_cmp(&b[0]) + .then(a[1].total_cmp(&b[1])) + .then(a[2].total_cmp(&b[2])) + }); + best.dedup(); + best +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &DasCmop9, points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 3]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/das_cmop9/output.txt b/examples/das_cmop9/output.txt new file mode 100644 index 00000000..b9841f6f --- /dev/null +++ b/examples/das_cmop9/output.txt @@ -0,0 +1,5 @@ +MOEA/D-DE, 1000 generations: 196 solutions, 196 feasible, IGD+ 0.0158, hypervolume 0.7484, 97.9% of the sample's; from the front: 0.0018 (median), 0.0200 (farthest) +NSGA-II, 1000 generations: 300 solutions, 300 feasible, IGD+ 0.2006, hypervolume 0.4880, 63.8% of the sample's; from the front: 0.0017 (median), 0.0471 (farthest) +the best point of the front for each of 300 weight vectors: 223 points, IGD+ 0.0184, hypervolume 0.7397, 96.7% of the sample's +the best point of the front for each of 666 weight vectors: 492 points, IGD+ 0.0121, hypervolume 0.7597, 99.4% of the sample's +the whole front: hypervolume 0.7853; a sample of 346 of its points: 0.7647 diff --git a/examples/das_cmop9/trace.json b/examples/das_cmop9/trace.json new file mode 100644 index 00000000..841a9638 --- /dev/null +++ b/examples/das_cmop9/trace.json @@ -0,0 +1,26 @@ 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+]} diff --git a/examples/das_cmop9/trace.py b/examples/das_cmop9/trace.py new file mode 100644 index 00000000..5aba0c8b --- /dev/null +++ b/examples/das_cmop9/trace.py @@ -0,0 +1,215 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: each run's population as the problem scores it, its feasible +non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +front's hypervolume with normalized objectives, in at most 24 generations; a frame's +evaluations are the runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the most infeasible solutions a frame shows +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions and the front's hypervolume + self.series = {} + + def front(self, name): + """The callback for ``run`` that records the run ``name`` after each generation, its + population scored by the problem, which the run may not optimize as it is: None without a + trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + objectives, violations = self.problem.evaluate(progress.population) + front = non_dominated(objectives[violations == 0]) + infeasible = objectives[violations > 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + history.append((progress.evaluations, front.tolist(), infeasible.tolist(), volume)) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(24) + for generation in range(max(map(len, self.series.values()), default=0)): + at = [runs[min(generation, len(runs) - 1)] for runs in self.series.values()] + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at), + "best": None, + "median": None, + "series": {name: record[3] for name, record in zip(self.series, at)}, + "state": { + "panels": [{"front": record[1], "infeasible": record[2]} for record in at] + }, + } + ) + # the front, sampled, for every panel + panel = { + "objectives": ["f1", "f2", "f3"], + "true_front": self.problem.optimal_front(600).tolist(), + } + panels = [{"title": name, "problem": panel} for name in self.series] + whole = self.normalized(self.problem.optimal_front(3_000)) + settings = { + "format": 1, + "example": "das_cmop9", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": list(self.series), + }, + } + write(self.path, settings, frames.to_list()) + + +def non_dominated(points): + """The points that no other point dominates, in their order.""" + keep = [ + not np.any(np.all(points <= point, axis=1) & np.any(points < point, axis=1)) + for point in points + ] + return points[np.array(keep, dtype=bool)] if len(points) else points + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/das_cmop9/trace.rs b/examples/das_cmop9/trace.rs new file mode 100644 index 00000000..3b2548e3 --- /dev/null +++ b/examples/das_cmop9/trace.rs @@ -0,0 +1,307 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: each run's population as the problem scores it, its feasible +//! non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +//! front's hypervolume with normalized objectives, in at most 24 generations; a frame's +//! evaluations are the runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::MultiSnapshot; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{DasCmop9, MultiProblem}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, and the +// hypervolume +type Record = (u64, Vec<[f64; 3]>, Vec<[f64; 3]>, f64); + +// the most infeasible solutions a frame shows +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation: its population scored by + // `problem`, which the run may not optimize as it is + pub fn front( + &mut self, + name: &'static str, + problem: DasCmop9, + ) -> impl FnMut(&MultiSnapshot<'_, Reals, 3>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let population = snapshot.population().as_slice(); + let scores: Vec<([f64; 3], f64)> = population + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = + scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let front = non_dominated(&feasible); + let infeasible = scores.iter().filter(|s| s.1 > 0.0).map(|s| s.0); + let infeasible: Vec<[f64; 3]> = infeasible.take(SHOWN).collect(); + let volume = hypervolume(&normalized(&problem, &front), &REFERENCE, &[Minimize; 3]); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &DasCmop9) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(24); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let (mut panels, mut volumes, mut evaluations) = (Vec::new(), Map::new(), 0); + for (name, history) in &self.series { + let (done, front, infeasible, volume) = &history[generation.min(history.len() - 1)]; + panels.push(json!({ "front": front, "infeasible": infeasible })); + volumes.insert(name.to_string(), json!(volume)); + evaluations += done; + } + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "series": volumes, + "state": { "panels": panels }, + })); + } + // the front, sampled, for every panel + let panel = json!({ + "objectives": ["f1", "f2", "f3"], + "true_front": problem.optimal_front(600).expect("known"), + }); + let panels: Vec = self + .series + .iter() + .map(|(name, _)| json!({ "title": name, "problem": panel })) + .collect(); + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(3_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "das_cmop9", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": names, + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the points that no other point dominates, in their order +fn non_dominated(points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let dominates = + |p: &[f64; 3], q: &[f64; 3]| (0..3).all(|j| p[j] <= q[j]) && (0..3).any(|j| p[j] < q[j]); + let kept = points + .iter() + .filter(|q| !points.iter().any(|p| dominates(p, q))); + kept.copied().collect() +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/dc1_dtlz1_3obj/README.md b/examples/dc1_dtlz1_3obj/README.md new file mode 100644 index 00000000..4d3b1966 --- /dev/null +++ b/examples/dc1_dtlz1_3obj/README.md @@ -0,0 +1,61 @@ +--- +title: DC1-DTLZ1 with 3 objectives +category: multi-objective +summary: Minimize three objectives of DTLZ1 subject to a constraint on the first variable, whose Pareto front is two bands of DTLZ1's, with NSGA-III and SMS-EMOA. +reference: "Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for constrained multiobjective optimization. IEEE Transactions on Evolutionary Computation 23(2): 303-315." +reference_url: https://doi.org/10.1109/TEVC.2018.2855411 +optimum: "two bands of the triangle f₁ + f₂ + f₃ = 1/2, f₃ in [1/9, 2/9] or [4/9, 1/2]; ideal point (0, 0, 1/9), nadir point (7/18, 7/18, 1/2); hypervolume 1.0709 (normalized objectives, reference point (1.1, 1.1, 1.1))" +languages: [rust, python] +order: 201 +family: DC-DTLZ +tab: DC1-DTLZ1 +--- + +# DC1-DTLZ1 with 3 objectives + +## The problem + +Li, Chen, Fu and Yao (2019) proposed C-TAEA, a two-archive algorithm for constrained problems, and, to test it, the DC-DTLZ problems: DTLZ1 and DTLZ3 with constraints on the decision variables, where Jain and Deb's C-DTLZ problems constrain the objectives. Type 1 (DC1) constrains the first variable and cuts the front into cones from the origin; type 2 (DC2) constrains the distance function g and leaves almost nothing but the front feasible, with local optima of the violation on the way; type 3 (DC3) does both, on every position variable. The paper's supplement defines them (section 1.2) with parameters a and b, and gives a = 3 and b = 0.5 for DC1 only; genoxide takes the others from the code of the authors' laboratory, EMOC, and the docs of `multi::problems::Dc1Dtlz1` say what else the supplement leaves open and how genoxide settles it. + +DC1-DTLZ1 with 3 objectives and 7 variables in [0, 1]: + +```text +minimize f₁ = ½ x₁ x₂ (1 + g) + f₂ = ½ x₁ (1 − x₂) (1 + g) + f₃ = ½ (1 − x₁) (1 + g) + g = 100 (5 + Σᵢ₌₃⁷ ((xᵢ − 0.5)² − cos(20π (xᵢ − 0.5)))) +subject to cos(aπx₁) ≥ b, a = 3, b = 0.5 +``` + +DTLZ1's front is the triangle where the objectives sum to 1/2, reached with the distance variables at 0.5, where g = 0. The constraint keeps x₁ in [0, 1/9] or [5/9, 7/9], a third of its range, and the front is the parts of the triangle with f₃ = (1 − x₁)/2 in [1/9, 2/9] or [4/9, 1/2]: two bands, each a cone from the origin through the triangle. Its ideal point is (0, 0, 1/9) and its nadir point (7/18, 7/18, 1/2). The supplement gives no front; genoxide derives it (g = 0 is feasible under every constraint, and the position constraints don't involve g, so DTLZ1's front points are optimal wherever their position variables are feasible, and no other point is), and it agrees with the sampled front that EMOC ships. + +## What makes it hard + +DTLZ1's g, with 11⁵ − 1 local minima, puts local fronts parallel to the true one, at g = 1, 2, …, which the population climbs down. The constraint cuts every one of them into the same cones, all feasible: the difficulty is to keep the population spread over both bands, when a third of the reference directions of NSGA-III point into the infeasible gaps between them. + +## Representation + +A `Real` genome of 7 genes in [0, 1]. The problem is genoxide's `Dc1Dtlz1`, whose fitness is the three objectives and the total constraint violation, 0 when it is feasible; in Python, `gx.problems.Dc1Dtlz1()`. Solutions compare by constrained dominance: a feasible solution beats an infeasible one, of two infeasible ones the smaller violation wins, and of two feasible ones Pareto dominance decides. + +## Algorithm + +Two runs, with a population of 92 for 1,000 generations each: + +- NSGA-III (Deb and Jain, 2014; Jain and Deb, 2014, IEEE Transactions on Evolutionary Computation 18(4): 577-601 and 602-622) with the settings that the C-TAEA paper gives its C-NSGA-III (supplement, tables 3 and 4): the 91 reference directions of Das and Dennis's method with 12 divisions and a population of 92, simulated binary crossover with η = 30 at a rate of 1, and polynomial mutation with η = 20 at a rate of 1/n per gene, for 1,000 generations (the supplement gives no budget for the DC-DTLZ problems; 2,000 on DTLZ3, whose local fronts take longer); +- SMS-EMOA (Beume, Naujoks and Emmerich, 2007, European Journal of Operational Research 181(3): 1653-1669), which keeps the solutions that add the most hypervolume, with the same operators and as many children a generation as the population. + +## Output + +A line per run: the size of its final front, how many of its solutions the problem finds feasible, their IGD+ and hypervolume, and the hypervolume as a share of that of a sample of the optimal front with at least as many points as the population. Then the hypervolumes of the whole front and of the sample. + +The indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 2,000 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest feasible solution of the found front, counting only the objectives in which the solution is worse. Smaller is better; a front of as many points as the population can't cover the 2,000 exactly, and the sample shows what it can. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the volume the feasible solutions dominate up to the reference point (1.1, 1.1, 1.1). Larger is better; the whole front's is computed from 3,000 of its points, and the sample's is what a front of that many points reaches. In Python, `run` evaluates the problem in Rust, so both versions print the same. + +The page plays the runs back side by side, each population as the problem scores it: its feasible non-dominated solutions, its infeasible ones (hollow, at most 100 a frame), and the optimal front, sampled. + +[The project page](https://tachsin.gr/projects/genoxide/examples/dc1-dtlz1-3obj) plays these runs back. + +## Good results + +The target: every solution feasible, and 99% of the hypervolume of the sample of 93 points of the front, about what a front of 92 solutions spread like it has. + +NSGA-III's run with seed 1 ends with 92 solutions, 92 feasible, IGD+ 0.0267, hypervolume 1.0434, 103.1% of the sample's; SMS-EMOA's with 92 solutions, 92 feasible, IGD+ 0.0247, hypervolume 1.0543, 104.2% of the sample's. Over seeds 1 to 20, NSGA-III ends with IGD+ from 0.0248 to 0.0271 and 102.8% to 103.4% of the sample's hypervolume, every run reaching the target, and SMS-EMOA with IGD+ from 0.0240 to 0.0253 and 104.1% to 104.3% of the sample's hypervolume, every run reaching the target. A front of 92 points spread over two bands can't come close to the 2,000 points that IGD+ measures with, and both exceed the sample's hypervolume: the solutions crowd the bands' edges, which add more volume. The C-TAEA paper's C-NSGA-III ends with a median IGD of 0.0599 (its table 3), and C-TAEA with 0.0564. diff --git a/examples/dc1_dtlz1_3obj/main.py b/examples/dc1_dtlz1_3obj/main.py new file mode 100644 index 00000000..52b103fa --- /dev/null +++ b/examples/dc1_dtlz1_3obj/main.py @@ -0,0 +1,94 @@ +"""DC1-DTLZ1 with 3 objectives: minimize three objectives of DTLZ1 subject to a constraint on the +first variable, whose front is two bands of DTLZ's, with NSGA-III and SMS-EMOA. + +From genoxide's problems.Dc1Dtlz1; run evaluates it in Rust. Runs NSGA-III with the settings of the +C-TAEA paper's C-NSGA-III and SMS-EMOA, a population of 92 for 1,000 generations each, with +constraint dominance. Prints each final front's size, how many of its solutions are feasible, their +IGD+ to 2,000 points of the optimal front and their hypervolume, with the objectives normalized by +the front's ideal and nadir points, as a share of that of a sample of the front with at least as +many points as the population; then the hypervolumes of the whole front and of the sample. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/dc1_dtlz1_3obj/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points +REFERENCE = [1.1, 1.1, 1.1] + +problem = gx.problems.Dc1Dtlz1() +ideal, nadir = problem.ideal_point, problem.nadir_point +rate = 1 / problem.dimensions + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) +# the sample: at least as many points of the optimal front as the population has, about what a +# front of that many solutions can be +sample = normalized(problem.optimal_front(92)) + + +def run(name, algorithm, fitness, generations): + """Runs ``algorithm`` on ``fitness`` for ``generations``, and prints its final front's size, + how many of its solutions the problem finds feasible, their IGD+ to 2,000 points of the + optimal front and their hypervolume, with normalized objectives, as a share of the + sample's.""" + result = algorithm.run(fitness, generations=generations, on_generation=trace.front(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"{name}, {generations} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(2_000)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + percent = 100 * volume / gx.indicators.hypervolume(sample, REFERENCE) + print( + start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}, " + f"{percent:.1f}% of the sample's" + ) + + +# NSGA-III with the settings of the C-TAEA paper's C-NSGA-III +algorithm = gx.Nsga3( + problem.genome, + objectives=problem.objectives, + reference_directions=gx.das_dennis(3, 12), + population_size=92, + crossover=gx.SimulatedBinaryCrossover(30), + crossover_rate=1.0, + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-III", algorithm, problem, 1_000) +# SMS-EMOA, which keeps the solutions that add the most hypervolume +algorithm = gx.SmsEmoa( + problem.genome, + objectives=problem.objectives, + population_size=92, + crossover=gx.SimulatedBinaryCrossover(30), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("SMS-EMOA", algorithm, problem, 1_000) +whole = normalized(problem.optimal_front(3_000)) +print( + f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}; " + f"a sample of {len(sample)} of its points: " + f"{gx.indicators.hypervolume(sample, REFERENCE):.4f}" +) +trace.write() diff --git a/examples/dc1_dtlz1_3obj/main.rs b/examples/dc1_dtlz1_3obj/main.rs new file mode 100644 index 00000000..f1600c1d --- /dev/null +++ b/examples/dc1_dtlz1_3obj/main.rs @@ -0,0 +1,134 @@ +//! DC1-DTLZ1 with 3 objectives: minimize three objectives of DTLZ1 subject to a constraint on the +//! first variable, whose front is two bands of DTLZ's, with NSGA-III and SMS-EMOA. +//! +//! From genoxide's `multi::problems::Dc1Dtlz1`. Runs NSGA-III with the settings of the C-TAEA +//! paper's C-NSGA-III and SMS-EMOA, a population of 92 for 1,000 generations each, with constraint +//! dominance. Prints each final front's size, how many of its solutions are feasible, their IGD+ to +//! 2,000 points of the optimal front and their hypervolume, with the objectives normalized by the +//! front's ideal and nadir points, as a share of that of a sample of the front with at least as +//! many points as the population; then the hypervolumes of the whole front and of the sample. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example dc1_dtlz1_3obj +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{Dc1Dtlz1, MultiProblem}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points +pub const REFERENCE: [f64; 3] = [1.1, 1.1, 1.1]; + +fn main() -> Result<()> { + let problem = Dc1Dtlz1::<3>::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the sample: at least as many points of the optimal front as the population has, about what + // a front of that many solutions can be + let sample = normalized(&problem, &problem.optimal_front(92).expect("known")); + let rate = 1.0 / problem.variables() as f64; + // NSGA-III with the settings of the C-TAEA paper's C-NSGA-III + let algorithm = Nsga3::builder( + problem.representation(), + [Minimize; 3], + multi::das_dennis::<3>(12), + ) + .population_size(92) + .crossover(SimulatedBinaryCrossover::new(30.0)?) + .crossover_rate(1.0) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, "NSGA-III", algorithm, problem, 1_000, &sample, &mut trace, + )?; + // SMS-EMOA, which keeps the solutions that add the most hypervolume + let algorithm = SmsEmoa::builder(problem.representation(), [Minimize; 3]) + .population_size(92) + .crossover(SimulatedBinaryCrossover::new(30.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, "SMS-EMOA", algorithm, problem, 1_000, &sample, &mut trace, + )?; + let whole = normalized(&problem, &problem.optimal_front(3_000).expect("known")); + println!( + "the whole front: hypervolume {:.4}; a sample of {} of its points: {:.4}", + hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + sample.len(), + hypervolume(&sample, &REFERENCE, &[Minimize; 3]) + ); + trace.write(&problem); + Ok(()) +} + +// runs `algorithm` on `fitness` for `generations`, and prints its final front's size, how many of +// its solutions the problem finds feasible, their IGD+ to 2,000 points of the optimal front and +// their hypervolume, with normalized objectives, as a share of the sample's +fn run( + problem: &Dc1Dtlz1<3>, + name: &'static str, + algorithm: A, + fitness: F, + generations: u64, + sample: &[[f64; 3]], + trace: &mut trace::Trace, +) -> Result<()> +where + A: MultiObjectiveAlgorithm<3, Genome = Reals>, + F: MultiFitnessFunction + Sync, +{ + let mut record = trace.front(name, *problem); + let outcome = MultiEngine::new(algorithm, fitness) + .stop_when(Stop::generations(generations)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 3], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "{name}, {generations} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(2_000).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 3]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 3]); + let percent = 100.0 * volume / hypervolume(sample, &REFERENCE, &[Minimize; 3]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the \ + sample's", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &Dc1Dtlz1<3>, points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 3]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/dc1_dtlz1_3obj/output.txt b/examples/dc1_dtlz1_3obj/output.txt new file mode 100644 index 00000000..5545f64a --- /dev/null +++ b/examples/dc1_dtlz1_3obj/output.txt @@ -0,0 +1,3 @@ +NSGA-III, 1000 generations: 92 solutions, 92 feasible, IGD+ 0.0267, hypervolume 1.0434, 103.1% of the sample's +SMS-EMOA, 1000 generations: 92 solutions, 92 feasible, IGD+ 0.0247, hypervolume 1.0543, 104.2% of the sample's +the whole front: hypervolume 1.0709; a sample of 93 of its points: 1.0118 diff --git a/examples/dc1_dtlz1_3obj/trace.json b/examples/dc1_dtlz1_3obj/trace.json new file mode 100644 index 00000000..6fb3264b --- /dev/null +++ b/examples/dc1_dtlz1_3obj/trace.json @@ -0,0 +1,42 @@ 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+]} diff --git a/examples/dc1_dtlz1_3obj/trace.py b/examples/dc1_dtlz1_3obj/trace.py new file mode 100644 index 00000000..4cc45ad0 --- /dev/null +++ b/examples/dc1_dtlz1_3obj/trace.py @@ -0,0 +1,215 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: each run's population as the problem scores it, its feasible +non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +front's hypervolume with normalized objectives, in at most 40 generations; a frame's +evaluations are the runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the most infeasible solutions a frame shows +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions and the front's hypervolume + self.series = {} + + def front(self, name): + """The callback for ``run`` that records the run ``name`` after each generation, its + population scored by the problem, which the run may not optimize as it is: None without a + trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + objectives, violations = self.problem.evaluate(progress.population) + front = non_dominated(objectives[violations == 0]) + infeasible = objectives[violations > 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + history.append((progress.evaluations, front.tolist(), infeasible.tolist(), volume)) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(40) + for generation in range(max(map(len, self.series.values()), default=0)): + at = [runs[min(generation, len(runs) - 1)] for runs in self.series.values()] + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at), + "best": None, + "median": None, + "series": {name: record[3] for name, record in zip(self.series, at)}, + "state": { + "panels": [{"front": record[1], "infeasible": record[2]} for record in at] + }, + } + ) + # the front, sampled, for every panel + panel = { + "objectives": ["f1", "f2", "f3"], + "true_front": self.problem.optimal_front(600).tolist(), + } + panels = [{"title": name, "problem": panel} for name in self.series] + whole = self.normalized(self.problem.optimal_front(3_000)) + settings = { + "format": 1, + "example": "dc1_dtlz1_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": list(self.series), + }, + } + write(self.path, settings, frames.to_list()) + + +def non_dominated(points): + """The points that no other point dominates, in their order.""" + keep = [ + not np.any(np.all(points <= point, axis=1) & np.any(points < point, axis=1)) + for point in points + ] + return points[np.array(keep, dtype=bool)] if len(points) else points + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/dc1_dtlz1_3obj/trace.rs b/examples/dc1_dtlz1_3obj/trace.rs new file mode 100644 index 00000000..a6e03e9c --- /dev/null +++ b/examples/dc1_dtlz1_3obj/trace.rs @@ -0,0 +1,307 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: each run's population as the problem scores it, its feasible +//! non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +//! front's hypervolume with normalized objectives, in at most 40 generations; a frame's +//! evaluations are the runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::MultiSnapshot; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{Dc1Dtlz1, MultiProblem}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, and the +// hypervolume +type Record = (u64, Vec<[f64; 3]>, Vec<[f64; 3]>, f64); + +// the most infeasible solutions a frame shows +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation: its population scored by + // `problem`, which the run may not optimize as it is + pub fn front( + &mut self, + name: &'static str, + problem: Dc1Dtlz1<3>, + ) -> impl FnMut(&MultiSnapshot<'_, Reals, 3>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let population = snapshot.population().as_slice(); + let scores: Vec<([f64; 3], f64)> = population + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = + scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let front = non_dominated(&feasible); + let infeasible = scores.iter().filter(|s| s.1 > 0.0).map(|s| s.0); + let infeasible: Vec<[f64; 3]> = infeasible.take(SHOWN).collect(); + let volume = hypervolume(&normalized(&problem, &front), &REFERENCE, &[Minimize; 3]); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &Dc1Dtlz1<3>) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(40); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let (mut panels, mut volumes, mut evaluations) = (Vec::new(), Map::new(), 0); + for (name, history) in &self.series { + let (done, front, infeasible, volume) = &history[generation.min(history.len() - 1)]; + panels.push(json!({ "front": front, "infeasible": infeasible })); + volumes.insert(name.to_string(), json!(volume)); + evaluations += done; + } + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "series": volumes, + "state": { "panels": panels }, + })); + } + // the front, sampled, for every panel + let panel = json!({ + "objectives": ["f1", "f2", "f3"], + "true_front": problem.optimal_front(600).expect("known"), + }); + let panels: Vec = self + .series + .iter() + .map(|(name, _)| json!({ "title": name, "problem": panel })) + .collect(); + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(3_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "dc1_dtlz1_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": names, + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the points that no other point dominates, in their order +fn non_dominated(points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let dominates = + |p: &[f64; 3], q: &[f64; 3]| (0..3).all(|j| p[j] <= q[j]) && (0..3).any(|j| p[j] < q[j]); + let kept = points + .iter() + .filter(|q| !points.iter().any(|p| dominates(p, q))); + kept.copied().collect() +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/dc1_dtlz3_3obj/README.md b/examples/dc1_dtlz3_3obj/README.md new file mode 100644 index 00000000..be0ec02a --- /dev/null +++ b/examples/dc1_dtlz3_3obj/README.md @@ -0,0 +1,61 @@ +--- +title: DC1-DTLZ3 with 3 objectives +category: multi-objective +summary: Minimize three objectives of DTLZ3 subject to a constraint on the first variable, whose Pareto front is two bands of the unit sphere, with NSGA-III and SMS-EMOA; NSGA-III reaches it in every run of 20. +reference: "Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for constrained multiobjective optimization. IEEE Transactions on Evolutionary Computation 23(2): 303-315." +reference_url: https://doi.org/10.1109/TEVC.2018.2855411 +optimum: "two bands of the unit sphere's octant, f₃ in [0, sin(π/18)] or [sin(5π/18), sin(7π/18)]; ideal point (0, 0, 0), nadir point (1, 1, sin(7π/18)) = (1, 1, 0.9397); hypervolume 0.6529 (normalized objectives, reference point (1.1, 1.1, 1.1))" +languages: [rust, python] +order: 202 +family: DC-DTLZ +tab: DC1-DTLZ3 +--- + +# DC1-DTLZ3 with 3 objectives + +## The problem + +Li, Chen, Fu and Yao (2019) proposed C-TAEA, a two-archive algorithm for constrained problems, and, to test it, the DC-DTLZ problems: DTLZ1 and DTLZ3 with constraints on the decision variables, where Jain and Deb's C-DTLZ problems constrain the objectives. Type 1 (DC1) constrains the first variable and cuts the front into cones from the origin; type 2 (DC2) constrains the distance function g and leaves almost nothing but the front feasible, with local optima of the violation on the way; type 3 (DC3) does both, on every position variable. The paper's supplement defines them (section 1.2) with parameters a and b, and gives a = 3 and b = 0.5 for DC1 only; genoxide takes the others from the code of the authors' laboratory, EMOC, and the docs of `multi::problems::Dc1Dtlz1` say what else the supplement leaves open and how genoxide settles it. + +DC1-DTLZ3 with 3 objectives and 12 variables in [0, 1]: + +```text +minimize f₁ = (1 + g) cos(πx₁/2) cos(πx₂/2) + f₂ = (1 + g) cos(πx₁/2) sin(πx₂/2) + f₃ = (1 + g) sin(πx₁/2) + g = 100 (10 + Σᵢ₌₃¹² ((xᵢ − 0.5)² − cos(20π (xᵢ − 0.5)))) +subject to cos(aπx₁) ≥ b, a = 3, b = 0.5 +``` + +DTLZ3's front is the unit sphere's octant, reached with the distance variables at 0.5, where g = 0. The constraint keeps x₁ in [0, 1/9] or [5/9, 7/9], and the front is the parts of the sphere with f₃ = sin(πx₁/2) in [0, sin(π/18)] or [sin(5π/18), sin(7π/18)]: a band along the base of the octant and one around its upper half. Its ideal point is (0, 0, 0) and its nadir point (1, 1, sin(7π/18)) = (1, 1, 0.9397). The supplement gives no front; genoxide derives it (g = 0 is feasible under every constraint, and the position constraints don't involve g, so DTLZ3's front points are optimal wherever their position variables are feasible, and no other point is), and it agrees with the sampled front that EMOC ships. + +## What makes it hard + +DTLZ3's g, with 3¹⁰ − 1 local fronts (Deb et al.), spheres of larger radii, which the population climbs down, all of them cut into the same feasible cones. Then the bands: the lower one, along the base, is narrow, a sixth of the octant's height, and NSGA-III's reference directions between the bands, which point into the infeasible gap, attach solutions at its edges. + +## Representation + +A `Real` genome of 12 genes in [0, 1]. The problem is genoxide's `Dc1Dtlz3`, whose fitness is the three objectives and the total constraint violation, 0 when it is feasible; in Python, `gx.problems.Dc1Dtlz3()`. Solutions compare by constrained dominance: a feasible solution beats an infeasible one, of two infeasible ones the smaller violation wins, and of two feasible ones Pareto dominance decides. + +## Algorithm + +Two runs, with a population of 92 for 4,000 generations each: + +- NSGA-III (Deb and Jain, 2014; Jain and Deb, 2014, IEEE Transactions on Evolutionary Computation 18(4): 577-601 and 602-622) with the settings that the C-TAEA paper gives its C-NSGA-III (supplement, tables 3 and 4): the 91 reference directions of Das and Dennis's method with 12 divisions and a population of 92, simulated binary crossover with η = 30 at a rate of 1, and polynomial mutation with η = 20 at a rate of 1/n per gene, for 4,000 generations (the supplement gives no budget for the DC-DTLZ problems; DTLZ3's local fronts take longer than DTLZ1's, and with 2,000, as on DC2-DTLZ3 and DC3-DTLZ3, 13 runs of 20 reach the target here and the rest come within 2% of it); +- SMS-EMOA (Beume, Naujoks and Emmerich, 2007, European Journal of Operational Research 181(3): 1653-1669), which keeps the solutions that add the most hypervolume, with the same operators and as many children a generation as the population. + +## Output + +A line per run: the size of its final front, how many of its solutions the problem finds feasible, their IGD+ and hypervolume, and the hypervolume as a share of that of a sample of the optimal front with at least as many points as the population. Then the hypervolumes of the whole front and of the sample. + +The indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 2,000 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest feasible solution of the found front, counting only the objectives in which the solution is worse. Smaller is better; a front of as many points as the population can't cover the 2,000 exactly, and the sample shows what it can. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the volume the feasible solutions dominate up to the reference point (1.1, 1.1, 1.1). Larger is better; the whole front's is computed from 3,000 of its points, and the sample's is what a front of that many points reaches. In Python, `run` evaluates the problem in Rust, so both versions print the same. + +The page plays the runs back side by side, each population as the problem scores it: its feasible non-dominated solutions, its infeasible ones (hollow, at most 100 a frame), and the optimal front, sampled. + +[The project page](https://tachsin.gr/projects/genoxide/examples/dc1-dtlz3-3obj) plays these runs back. + +## Good results + +The target: every solution feasible, and 99% of the hypervolume of the sample of 100 points of the front, about what a front of 92 solutions spread like it has. + +NSGA-III's run with seed 1 ends with 92 solutions, 92 feasible, IGD+ 0.0170, hypervolume 0.6236, 99.4% of the sample's; SMS-EMOA's with 92 solutions, 92 feasible, IGD+ 0.0507, hypervolume 0.5268, 84.0% of the sample's. Over seeds 1 to 20, NSGA-III ends with IGD+ from 0.0156 to 0.0174 and 99.2% to 99.7% of the sample's hypervolume, every run reaching the target, and SMS-EMOA with IGD+ from 0.0501 to 0.0583 and 79.4% to 84.3% of the sample's hypervolume, no run reaching the target. NSGA-III needs the 4,000 generations: after 2,000, 13 runs reach the target and the rest are within 2% of it, the directions that point into the gap between the bands spending solutions on its edges. SMS-EMOA, whose hypervolume selection favours the upper band, leaves the lower one thin. The C-TAEA paper's C-NSGA-III ends with a median IGD of 0.272 (its table 3), and C-TAEA with 0.147. diff --git a/examples/dc1_dtlz3_3obj/main.py b/examples/dc1_dtlz3_3obj/main.py new file mode 100644 index 00000000..1aecb871 --- /dev/null +++ b/examples/dc1_dtlz3_3obj/main.py @@ -0,0 +1,94 @@ +"""DC1-DTLZ3 with 3 objectives: minimize three objectives of DTLZ3 subject to a constraint on the +first variable, whose front is two bands of DTLZ's, with NSGA-III and SMS-EMOA. + +From genoxide's problems.Dc1Dtlz3; run evaluates it in Rust. Runs NSGA-III with the settings of the +C-TAEA paper's C-NSGA-III and SMS-EMOA, a population of 92 for 4,000 generations each, with +constraint dominance. Prints each final front's size, how many of its solutions are feasible, their +IGD+ to 2,000 points of the optimal front and their hypervolume, with the objectives normalized by +the front's ideal and nadir points, as a share of that of a sample of the front with at least as +many points as the population; then the hypervolumes of the whole front and of the sample. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/dc1_dtlz3_3obj/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points +REFERENCE = [1.1, 1.1, 1.1] + +problem = gx.problems.Dc1Dtlz3() +ideal, nadir = problem.ideal_point, problem.nadir_point +rate = 1 / problem.dimensions + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) +# the sample: at least as many points of the optimal front as the population has, about what a +# front of that many solutions can be +sample = normalized(problem.optimal_front(92)) + + +def run(name, algorithm, fitness, generations): + """Runs ``algorithm`` on ``fitness`` for ``generations``, and prints its final front's size, + how many of its solutions the problem finds feasible, their IGD+ to 2,000 points of the + optimal front and their hypervolume, with normalized objectives, as a share of the + sample's.""" + result = algorithm.run(fitness, generations=generations, on_generation=trace.front(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"{name}, {generations} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(2_000)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + percent = 100 * volume / gx.indicators.hypervolume(sample, REFERENCE) + print( + start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}, " + f"{percent:.1f}% of the sample's" + ) + + +# NSGA-III with the settings of the C-TAEA paper's C-NSGA-III +algorithm = gx.Nsga3( + problem.genome, + objectives=problem.objectives, + reference_directions=gx.das_dennis(3, 12), + population_size=92, + crossover=gx.SimulatedBinaryCrossover(30), + crossover_rate=1.0, + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-III", algorithm, problem, 4_000) +# SMS-EMOA, which keeps the solutions that add the most hypervolume +algorithm = gx.SmsEmoa( + problem.genome, + objectives=problem.objectives, + population_size=92, + crossover=gx.SimulatedBinaryCrossover(30), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("SMS-EMOA", algorithm, problem, 4_000) +whole = normalized(problem.optimal_front(3_000)) +print( + f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}; " + f"a sample of {len(sample)} of its points: " + f"{gx.indicators.hypervolume(sample, REFERENCE):.4f}" +) +trace.write() diff --git a/examples/dc1_dtlz3_3obj/main.rs b/examples/dc1_dtlz3_3obj/main.rs new file mode 100644 index 00000000..429b676d --- /dev/null +++ b/examples/dc1_dtlz3_3obj/main.rs @@ -0,0 +1,134 @@ +//! DC1-DTLZ3 with 3 objectives: minimize three objectives of DTLZ3 subject to a constraint on the +//! first variable, whose front is two bands of DTLZ's, with NSGA-III and SMS-EMOA. +//! +//! From genoxide's `multi::problems::Dc1Dtlz3`. Runs NSGA-III with the settings of the C-TAEA +//! paper's C-NSGA-III and SMS-EMOA, a population of 92 for 4,000 generations each, with constraint +//! dominance. Prints each final front's size, how many of its solutions are feasible, their IGD+ to +//! 2,000 points of the optimal front and their hypervolume, with the objectives normalized by the +//! front's ideal and nadir points, as a share of that of a sample of the front with at least as +//! many points as the population; then the hypervolumes of the whole front and of the sample. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example dc1_dtlz3_3obj +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{Dc1Dtlz3, MultiProblem}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points +pub const REFERENCE: [f64; 3] = [1.1, 1.1, 1.1]; + +fn main() -> Result<()> { + let problem = Dc1Dtlz3::<3>::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the sample: at least as many points of the optimal front as the population has, about what + // a front of that many solutions can be + let sample = normalized(&problem, &problem.optimal_front(92).expect("known")); + let rate = 1.0 / problem.variables() as f64; + // NSGA-III with the settings of the C-TAEA paper's C-NSGA-III + let algorithm = Nsga3::builder( + problem.representation(), + [Minimize; 3], + multi::das_dennis::<3>(12), + ) + .population_size(92) + .crossover(SimulatedBinaryCrossover::new(30.0)?) + .crossover_rate(1.0) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, "NSGA-III", algorithm, problem, 4_000, &sample, &mut trace, + )?; + // SMS-EMOA, which keeps the solutions that add the most hypervolume + let algorithm = SmsEmoa::builder(problem.representation(), [Minimize; 3]) + .population_size(92) + .crossover(SimulatedBinaryCrossover::new(30.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, "SMS-EMOA", algorithm, problem, 4_000, &sample, &mut trace, + )?; + let whole = normalized(&problem, &problem.optimal_front(3_000).expect("known")); + println!( + "the whole front: hypervolume {:.4}; a sample of {} of its points: {:.4}", + hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + sample.len(), + hypervolume(&sample, &REFERENCE, &[Minimize; 3]) + ); + trace.write(&problem); + Ok(()) +} + +// runs `algorithm` on `fitness` for `generations`, and prints its final front's size, how many of +// its solutions the problem finds feasible, their IGD+ to 2,000 points of the optimal front and +// their hypervolume, with normalized objectives, as a share of the sample's +fn run( + problem: &Dc1Dtlz3<3>, + name: &'static str, + algorithm: A, + fitness: F, + generations: u64, + sample: &[[f64; 3]], + trace: &mut trace::Trace, +) -> Result<()> +where + A: MultiObjectiveAlgorithm<3, Genome = Reals>, + F: MultiFitnessFunction + Sync, +{ + let mut record = trace.front(name, *problem); + let outcome = MultiEngine::new(algorithm, fitness) + .stop_when(Stop::generations(generations)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 3], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "{name}, {generations} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(2_000).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 3]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 3]); + let percent = 100.0 * volume / hypervolume(sample, &REFERENCE, &[Minimize; 3]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the \ + sample's", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &Dc1Dtlz3<3>, points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 3]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/dc1_dtlz3_3obj/output.txt b/examples/dc1_dtlz3_3obj/output.txt new file mode 100644 index 00000000..e575209a --- /dev/null +++ b/examples/dc1_dtlz3_3obj/output.txt @@ -0,0 +1,3 @@ +NSGA-III, 4000 generations: 92 solutions, 92 feasible, IGD+ 0.0170, hypervolume 0.6236, 99.4% of the sample's +SMS-EMOA, 4000 generations: 92 solutions, 92 feasible, IGD+ 0.0507, hypervolume 0.5268, 84.0% of the sample's +the whole front: hypervolume 0.6529; a sample of 100 of its points: 0.6273 diff --git a/examples/dc1_dtlz3_3obj/trace.json b/examples/dc1_dtlz3_3obj/trace.json new file mode 100644 index 00000000..eb26b9fd --- /dev/null +++ b/examples/dc1_dtlz3_3obj/trace.json @@ -0,0 +1,42 @@ 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+]} diff --git a/examples/dc1_dtlz3_3obj/trace.py b/examples/dc1_dtlz3_3obj/trace.py new file mode 100644 index 00000000..07b13f45 --- /dev/null +++ b/examples/dc1_dtlz3_3obj/trace.py @@ -0,0 +1,215 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: each run's population as the problem scores it, its feasible +non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +front's hypervolume with normalized objectives, in at most 40 generations; a frame's +evaluations are the runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the most infeasible solutions a frame shows +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions and the front's hypervolume + self.series = {} + + def front(self, name): + """The callback for ``run`` that records the run ``name`` after each generation, its + population scored by the problem, which the run may not optimize as it is: None without a + trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + objectives, violations = self.problem.evaluate(progress.population) + front = non_dominated(objectives[violations == 0]) + infeasible = objectives[violations > 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + history.append((progress.evaluations, front.tolist(), infeasible.tolist(), volume)) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(40) + for generation in range(max(map(len, self.series.values()), default=0)): + at = [runs[min(generation, len(runs) - 1)] for runs in self.series.values()] + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at), + "best": None, + "median": None, + "series": {name: record[3] for name, record in zip(self.series, at)}, + "state": { + "panels": [{"front": record[1], "infeasible": record[2]} for record in at] + }, + } + ) + # the front, sampled, for every panel + panel = { + "objectives": ["f1", "f2", "f3"], + "true_front": self.problem.optimal_front(600).tolist(), + } + panels = [{"title": name, "problem": panel} for name in self.series] + whole = self.normalized(self.problem.optimal_front(3_000)) + settings = { + "format": 1, + "example": "dc1_dtlz3_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": list(self.series), + }, + } + write(self.path, settings, frames.to_list()) + + +def non_dominated(points): + """The points that no other point dominates, in their order.""" + keep = [ + not np.any(np.all(points <= point, axis=1) & np.any(points < point, axis=1)) + for point in points + ] + return points[np.array(keep, dtype=bool)] if len(points) else points + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/dc1_dtlz3_3obj/trace.rs b/examples/dc1_dtlz3_3obj/trace.rs new file mode 100644 index 00000000..d1ae79f5 --- /dev/null +++ b/examples/dc1_dtlz3_3obj/trace.rs @@ -0,0 +1,307 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: each run's population as the problem scores it, its feasible +//! non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +//! front's hypervolume with normalized objectives, in at most 40 generations; a frame's +//! evaluations are the runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::MultiSnapshot; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{Dc1Dtlz3, MultiProblem}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, and the +// hypervolume +type Record = (u64, Vec<[f64; 3]>, Vec<[f64; 3]>, f64); + +// the most infeasible solutions a frame shows +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation: its population scored by + // `problem`, which the run may not optimize as it is + pub fn front( + &mut self, + name: &'static str, + problem: Dc1Dtlz3<3>, + ) -> impl FnMut(&MultiSnapshot<'_, Reals, 3>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let population = snapshot.population().as_slice(); + let scores: Vec<([f64; 3], f64)> = population + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = + scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let front = non_dominated(&feasible); + let infeasible = scores.iter().filter(|s| s.1 > 0.0).map(|s| s.0); + let infeasible: Vec<[f64; 3]> = infeasible.take(SHOWN).collect(); + let volume = hypervolume(&normalized(&problem, &front), &REFERENCE, &[Minimize; 3]); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &Dc1Dtlz3<3>) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(40); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let (mut panels, mut volumes, mut evaluations) = (Vec::new(), Map::new(), 0); + for (name, history) in &self.series { + let (done, front, infeasible, volume) = &history[generation.min(history.len() - 1)]; + panels.push(json!({ "front": front, "infeasible": infeasible })); + volumes.insert(name.to_string(), json!(volume)); + evaluations += done; + } + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "series": volumes, + "state": { "panels": panels }, + })); + } + // the front, sampled, for every panel + let panel = json!({ + "objectives": ["f1", "f2", "f3"], + "true_front": problem.optimal_front(600).expect("known"), + }); + let panels: Vec = self + .series + .iter() + .map(|(name, _)| json!({ "title": name, "problem": panel })) + .collect(); + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(3_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "dc1_dtlz3_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": names, + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the points that no other point dominates, in their order +fn non_dominated(points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let dominates = + |p: &[f64; 3], q: &[f64; 3]| (0..3).all(|j| p[j] <= q[j]) && (0..3).any(|j| p[j] < q[j]); + let kept = points + .iter() + .filter(|q| !points.iter().any(|p| dominates(p, q))); + kept.copied().collect() +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/dc2_dtlz1_3obj/README.md b/examples/dc2_dtlz1_3obj/README.md new file mode 100644 index 00000000..7d5e5e1f --- /dev/null +++ b/examples/dc2_dtlz1_3obj/README.md @@ -0,0 +1,62 @@ +--- +title: DC2-DTLZ1 with 3 objectives +category: multi-objective +summary: Minimize three objectives of DTLZ1 subject to two constraints on its distance function, which leave almost nothing but the front feasible, with NSGA-III; constrained dominance stalls at a local minimum of the violation, and solving DTLZ without the constraints reaches the front. +reference: "Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for constrained multiobjective optimization. IEEE Transactions on Evolutionary Computation 23(2): 303-315." +reference_url: https://doi.org/10.1109/TEVC.2018.2855411 +optimum: "DTLZ1's front, whole; ideal point (0, 0, 0), nadir point (1/2, 1/2, 1/2); hypervolume 1.1577 (normalized objectives, reference point (1.1, 1.1, 1.1))" +languages: [rust, python] +order: 203 +family: DC-DTLZ +tab: DC2-DTLZ1 +--- + +# DC2-DTLZ1 with 3 objectives + +## The problem + +Li, Chen, Fu and Yao (2019) proposed C-TAEA, a two-archive algorithm for constrained problems, and, to test it, the DC-DTLZ problems: DTLZ1 and DTLZ3 with constraints on the decision variables, where Jain and Deb's C-DTLZ problems constrain the objectives. Type 1 (DC1) constrains the first variable and cuts the front into cones from the origin; type 2 (DC2) constrains the distance function g and leaves almost nothing but the front feasible, with local optima of the violation on the way; type 3 (DC3) does both, on every position variable. The paper's supplement defines them (section 1.2) with parameters a and b, and gives a = 3 and b = 0.5 for DC1 only; genoxide takes the others from the code of the authors' laboratory, EMOC, and the docs of `multi::problems::Dc1Dtlz1` say what else the supplement leaves open and how genoxide settles it. + +DC2-DTLZ1 with 3 objectives and 7 variables in [0, 1]: + +```text +minimize f₁ = ½ x₁ x₂ (1 + g) + f₂ = ½ x₁ (1 − x₂) (1 + g) + f₃ = ½ (1 − x₁) (1 + g) + g = 100 (5 + Σᵢ₌₃⁷ ((xᵢ − 0.5)² − cos(20π (xᵢ − 0.5)))) +subject to cos(aπg) ≥ b + e^(−g) ≥ b, a = 3, b = 0.9 +``` + +DTLZ1's front is the triangle where the objectives sum to 1/2, reached with the distance variables at 0.5, where g = 0. With b = 0.9 the constraints ask g ≤ 0.048, in the first band of the cosine (the second constraint asks g ≤ −ln 0.9 ≈ 0.105): feasible solutions lie within a thin shell of the front, and the front is DTLZ1's, whole. Its ideal point is (0, 0, 0) and its nadir point (1/2, 1/2, 1/2). The supplement gives no front; genoxide derives it (g = 0 is feasible under every constraint, and the position constraints don't involve g, so DTLZ1's front points are optimal wherever their position variables are feasible, and no other point is), and it agrees with the sampled front that EMOC ships. + +## What makes it hard + +The violation: b − cos(aπg) rises and falls with g, back to its least, b − 1, at g = 2/3, 4/3, …, where e^(−g) is the only constraint still broken. Constrained dominance ranks infeasible solutions by their violation alone, so a population that reaches one of these local minima of the violation stays there: to go on, it must climb out first. Almost nothing but the front is feasible, so it never finds a feasible solution to pull it down. + +## Representation + +A `Real` genome of 7 genes in [0, 1]. The problem is genoxide's `Dc2Dtlz1`, whose fitness is the three objectives and the total constraint violation, 0 when it is feasible; in Python, `gx.problems.Dc2Dtlz1()`. Solutions compare by constrained dominance: a feasible solution beats an infeasible one, of two infeasible ones the smaller violation wins, and of two feasible ones Pareto dominance decides. + +## Algorithm + +Two runs of NSGA-III, with a population of 92 for 1,000 generations each: + +- NSGA-III (Deb and Jain, 2014; Jain and Deb, 2014, IEEE Transactions on Evolutionary Computation 18(4): 577-601 and 602-622) with the settings that the C-TAEA paper gives its C-NSGA-III (supplement, tables 3 and 4): the 91 reference directions of Das and Dennis's method with 12 divisions and a population of 92, simulated binary crossover with η = 30 at a rate of 1, and polynomial mutation with η = 20 at a rate of 1/n per gene, for 1,000 generations (the supplement gives no budget for the DC-DTLZ problems; 2,000 on DTLZ3, whose local fronts take longer), with constrained dominance; +- the same on DTLZ1 itself, without the constraints, its final front then scored by DC2-DTLZ1: the front is DTLZ1's, and a population that converges to it is feasible. + +## Output + +A line per run: the size of its final front, how many of its solutions the problem finds feasible, their IGD+ and hypervolume, and the hypervolume as a share of that of a sample of the optimal front with at least as many points as the population. Then the hypervolumes of the whole front and of the sample. + +The indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 2,000 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest feasible solution of the found front, counting only the objectives in which the solution is worse. Smaller is better; a front of as many points as the population can't cover the 2,000 exactly, and the sample shows what it can. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the volume the feasible solutions dominate up to the reference point (1.1, 1.1, 1.1). Larger is better; the whole front's is computed from 3,000 of its points, and the sample's is what a front of that many points reaches. In Python, `run` evaluates the problem in Rust, so both versions print the same. + +The page plays the runs back side by side, each population as the problem scores it: its feasible non-dominated solutions, its infeasible ones (hollow, at most 100 a frame), and the optimal front, sampled. + +[The project page](https://tachsin.gr/projects/genoxide/examples/dc2-dtlz1-3obj) plays these runs back. + +## Good results + +The target: every solution feasible, and 99% of the hypervolume of the sample of 105 points of the front, about what a front of 92 solutions spread like it has. + +The run with constrained dominance ends with 1 solution, none feasible, the least violation 0.8975: stuck at a local minimum of the violation. Without the constraints, it ends with 92 solutions, 92 feasible, IGD+ 0.0289, hypervolume 1.1203, 99.7% of the sample's. Over seeds 1 to 20, every run with constrained dominance ends with no feasible solution, and without the constraints IGD+ from 0.0288 to 0.0307 and 99.4% to 99.7% of the sample's hypervolume, every run reaching the target. The C-TAEA paper's C-NSGA-III ends with a median IGD of 17.6 (its table 3), the same failure, and C-TAEA with 0.0220. Ignoring the constraints works here because they leave the front whole, as Tanabe and Oyama (2017) note of C1-DTLZ1 and C1-DTLZ3. diff --git a/examples/dc2_dtlz1_3obj/main.py b/examples/dc2_dtlz1_3obj/main.py new file mode 100644 index 00000000..aeca43d6 --- /dev/null +++ b/examples/dc2_dtlz1_3obj/main.py @@ -0,0 +1,97 @@ +"""DC2-DTLZ1 with 3 objectives: minimize three objectives of DTLZ1 subject to two constraints on its +distance function, which leave almost nothing but the front feasible, with NSGA-III. + +From genoxide's problems.Dc2Dtlz1; run evaluates it in Rust. Runs NSGA-III with the settings of the +C-TAEA paper's C-NSGA-III, a population of 92 for 1,000 generations, twice: with constraint +dominance, and on DTLZ1 without the constraints. Prints each final front's size, how many of its +solutions are feasible, their IGD+ to 2,000 points of the optimal front and their hypervolume, with +the objectives normalized by the front's ideal and nadir points, as a share of that of a sample of +the front with at least as many points as the population; then the hypervolumes of the whole front +and of the sample. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/dc2_dtlz1_3obj/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points +REFERENCE = [1.1, 1.1, 1.1] + +problem = gx.problems.Dc2Dtlz1() +ideal, nadir = problem.ideal_point, problem.nadir_point +rate = 1 / problem.dimensions + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) +# the sample: at least as many points of the optimal front as the population has, about what a +# front of that many solutions can be +sample = normalized(problem.optimal_front(92)) + + +def run(name, algorithm, fitness, generations): + """Runs ``algorithm`` on ``fitness`` for ``generations``, and prints its final front's size, + how many of its solutions the problem finds feasible, their IGD+ to 2,000 points of the + optimal front and their hypervolume, with normalized objectives, as a share of the + sample's.""" + result = algorithm.run(fitness, generations=generations, on_generation=trace.front(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"{name}, {generations} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(2_000)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + percent = 100 * volume / gx.indicators.hypervolume(sample, REFERENCE) + print( + start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}, " + f"{percent:.1f}% of the sample's" + ) + + +# NSGA-III with the settings of the C-TAEA paper's C-NSGA-III, constraint dominance +algorithm = gx.Nsga3( + problem.genome, + objectives=problem.objectives, + reference_directions=gx.das_dennis(3, 12), + population_size=92, + crossover=gx.SimulatedBinaryCrossover(30), + crossover_rate=1.0, + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-III", algorithm, problem, 1_000) +# the same on DTLZ1, without the constraints, whose solutions the problem then scores +algorithm = gx.Nsga3( + problem.genome, + objectives=problem.objectives, + reference_directions=gx.das_dennis(3, 12), + population_size=92, + crossover=gx.SimulatedBinaryCrossover(30), + crossover_rate=1.0, + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-III, unconstrained", algorithm, gx.problems.Dtlz1(variables=problem.dimensions), 1_000) +whole = normalized(problem.optimal_front(3_000)) +print( + f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}; " + f"a sample of {len(sample)} of its points: " + f"{gx.indicators.hypervolume(sample, REFERENCE):.4f}" +) +trace.write() diff --git a/examples/dc2_dtlz1_3obj/main.rs b/examples/dc2_dtlz1_3obj/main.rs new file mode 100644 index 00000000..879d1424 --- /dev/null +++ b/examples/dc2_dtlz1_3obj/main.rs @@ -0,0 +1,146 @@ +//! DC2-DTLZ1 with 3 objectives: minimize three objectives of DTLZ1 subject to two constraints on +//! its distance function, which leave almost nothing but the front feasible, with NSGA-III. +//! +//! From genoxide's `multi::problems::Dc2Dtlz1`. Runs NSGA-III with the settings of the C-TAEA +//! paper's C-NSGA-III, a population of 92 for 1,000 generations, twice: with constraint dominance, +//! and on DTLZ1 without the constraints. Prints each final front's size, how many of its solutions +//! are feasible, their IGD+ to 2,000 points of the optimal front and their hypervolume, with the +//! objectives normalized by the front's ideal and nadir points, as a share of that of a sample of +//! the front with at least as many points as the population; then the hypervolumes of the whole +//! front and of the sample. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example dc2_dtlz1_3obj +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{Dc2Dtlz1, Dtlz1, MultiProblem}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points +pub const REFERENCE: [f64; 3] = [1.1, 1.1, 1.1]; + +fn main() -> Result<()> { + let problem = Dc2Dtlz1::<3>::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the sample: at least as many points of the optimal front as the population has, about what + // a front of that many solutions can be + let sample = normalized(&problem, &problem.optimal_front(92).expect("known")); + let rate = 1.0 / problem.variables() as f64; + // NSGA-III with the settings of the C-TAEA paper's C-NSGA-III, constraint dominance + let algorithm = Nsga3::builder( + problem.representation(), + [Minimize; 3], + multi::das_dennis::<3>(12), + ) + .population_size(92) + .crossover(SimulatedBinaryCrossover::new(30.0)?) + .crossover_rate(1.0) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, "NSGA-III", algorithm, problem, 1_000, &sample, &mut trace, + )?; + // the same on DTLZ1, without the constraints, whose solutions the problem then scores + let algorithm = Nsga3::builder( + problem.representation(), + [Minimize; 3], + multi::das_dennis::<3>(12), + ) + .population_size(92) + .crossover(SimulatedBinaryCrossover::new(30.0)?) + .crossover_rate(1.0) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, + "NSGA-III, unconstrained", + algorithm, + Dtlz1::<3>::new(problem.variables()), + 1_000, + &sample, + &mut trace, + )?; + let whole = normalized(&problem, &problem.optimal_front(3_000).expect("known")); + println!( + "the whole front: hypervolume {:.4}; a sample of {} of its points: {:.4}", + hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + sample.len(), + hypervolume(&sample, &REFERENCE, &[Minimize; 3]) + ); + trace.write(&problem); + Ok(()) +} + +// runs `algorithm` on `fitness` for `generations`, and prints its final front's size, how many of +// its solutions the problem finds feasible, their IGD+ to 2,000 points of the optimal front and +// their hypervolume, with normalized objectives, as a share of the sample's +fn run( + problem: &Dc2Dtlz1<3>, + name: &'static str, + algorithm: A, + fitness: F, + generations: u64, + sample: &[[f64; 3]], + trace: &mut trace::Trace, +) -> Result<()> +where + A: MultiObjectiveAlgorithm<3, Genome = Reals>, + F: MultiFitnessFunction + Sync, +{ + let mut record = trace.front(name, *problem); + let outcome = MultiEngine::new(algorithm, fitness) + .stop_when(Stop::generations(generations)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 3], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "{name}, {generations} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(2_000).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 3]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 3]); + let percent = 100.0 * volume / hypervolume(sample, &REFERENCE, &[Minimize; 3]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the \ + sample's", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &Dc2Dtlz1<3>, points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 3]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/dc2_dtlz1_3obj/output.txt b/examples/dc2_dtlz1_3obj/output.txt new file mode 100644 index 00000000..35e94606 --- /dev/null +++ b/examples/dc2_dtlz1_3obj/output.txt @@ -0,0 +1,3 @@ +NSGA-III, 1000 generations: 1 solution, none feasible, the least violation 0.8975 +NSGA-III, unconstrained, 1000 generations: 92 solutions, 92 feasible, IGD+ 0.0289, hypervolume 1.1203, 99.7% of the sample's +the whole front: hypervolume 1.1577; a sample of 105 of its points: 1.1239 diff --git a/examples/dc2_dtlz1_3obj/trace.json b/examples/dc2_dtlz1_3obj/trace.json new file mode 100644 index 00000000..3dd9a0ca --- /dev/null +++ b/examples/dc2_dtlz1_3obj/trace.json @@ -0,0 +1,42 @@ 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+]} diff --git a/examples/dc2_dtlz1_3obj/trace.py b/examples/dc2_dtlz1_3obj/trace.py new file mode 100644 index 00000000..8dc4ce50 --- /dev/null +++ b/examples/dc2_dtlz1_3obj/trace.py @@ -0,0 +1,215 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: each run's population as the problem scores it, its feasible +non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +front's hypervolume with normalized objectives, in at most 40 generations; a frame's +evaluations are the runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the most infeasible solutions a frame shows +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions and the front's hypervolume + self.series = {} + + def front(self, name): + """The callback for ``run`` that records the run ``name`` after each generation, its + population scored by the problem, which the run may not optimize as it is: None without a + trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + objectives, violations = self.problem.evaluate(progress.population) + front = non_dominated(objectives[violations == 0]) + infeasible = objectives[violations > 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + history.append((progress.evaluations, front.tolist(), infeasible.tolist(), volume)) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(40) + for generation in range(max(map(len, self.series.values()), default=0)): + at = [runs[min(generation, len(runs) - 1)] for runs in self.series.values()] + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at), + "best": None, + "median": None, + "series": {name: record[3] for name, record in zip(self.series, at)}, + "state": { + "panels": [{"front": record[1], "infeasible": record[2]} for record in at] + }, + } + ) + # the front, sampled, for every panel + panel = { + "objectives": ["f1", "f2", "f3"], + "true_front": self.problem.optimal_front(600).tolist(), + } + panels = [{"title": name, "problem": panel} for name in self.series] + whole = self.normalized(self.problem.optimal_front(3_000)) + settings = { + "format": 1, + "example": "dc2_dtlz1_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": list(self.series), + }, + } + write(self.path, settings, frames.to_list()) + + +def non_dominated(points): + """The points that no other point dominates, in their order.""" + keep = [ + not np.any(np.all(points <= point, axis=1) & np.any(points < point, axis=1)) + for point in points + ] + return points[np.array(keep, dtype=bool)] if len(points) else points + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/dc2_dtlz1_3obj/trace.rs b/examples/dc2_dtlz1_3obj/trace.rs new file mode 100644 index 00000000..a519d0ef --- /dev/null +++ b/examples/dc2_dtlz1_3obj/trace.rs @@ -0,0 +1,307 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: each run's population as the problem scores it, its feasible +//! non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +//! front's hypervolume with normalized objectives, in at most 40 generations; a frame's +//! evaluations are the runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::MultiSnapshot; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{Dc2Dtlz1, MultiProblem}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, and the +// hypervolume +type Record = (u64, Vec<[f64; 3]>, Vec<[f64; 3]>, f64); + +// the most infeasible solutions a frame shows +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation: its population scored by + // `problem`, which the run may not optimize as it is + pub fn front( + &mut self, + name: &'static str, + problem: Dc2Dtlz1<3>, + ) -> impl FnMut(&MultiSnapshot<'_, Reals, 3>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let population = snapshot.population().as_slice(); + let scores: Vec<([f64; 3], f64)> = population + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = + scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let front = non_dominated(&feasible); + let infeasible = scores.iter().filter(|s| s.1 > 0.0).map(|s| s.0); + let infeasible: Vec<[f64; 3]> = infeasible.take(SHOWN).collect(); + let volume = hypervolume(&normalized(&problem, &front), &REFERENCE, &[Minimize; 3]); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &Dc2Dtlz1<3>) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(40); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let (mut panels, mut volumes, mut evaluations) = (Vec::new(), Map::new(), 0); + for (name, history) in &self.series { + let (done, front, infeasible, volume) = &history[generation.min(history.len() - 1)]; + panels.push(json!({ "front": front, "infeasible": infeasible })); + volumes.insert(name.to_string(), json!(volume)); + evaluations += done; + } + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "series": volumes, + "state": { "panels": panels }, + })); + } + // the front, sampled, for every panel + let panel = json!({ + "objectives": ["f1", "f2", "f3"], + "true_front": problem.optimal_front(600).expect("known"), + }); + let panels: Vec = self + .series + .iter() + .map(|(name, _)| json!({ "title": name, "problem": panel })) + .collect(); + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(3_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "dc2_dtlz1_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": names, + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the points that no other point dominates, in their order +fn non_dominated(points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let dominates = + |p: &[f64; 3], q: &[f64; 3]| (0..3).all(|j| p[j] <= q[j]) && (0..3).any(|j| p[j] < q[j]); + let kept = points + .iter() + .filter(|q| !points.iter().any(|p| dominates(p, q))); + kept.copied().collect() +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/dc2_dtlz3_3obj/README.md b/examples/dc2_dtlz3_3obj/README.md new file mode 100644 index 00000000..fa06248b --- /dev/null +++ b/examples/dc2_dtlz3_3obj/README.md @@ -0,0 +1,62 @@ +--- +title: DC2-DTLZ3 with 3 objectives +category: multi-objective +summary: Minimize three objectives of DTLZ3 subject to two constraints on its distance function, which leave almost nothing but the front feasible, with NSGA-III; constrained dominance stalls at a local minimum of the violation, and solving DTLZ without the constraints reaches the front. +reference: "Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for constrained multiobjective optimization. IEEE Transactions on Evolutionary Computation 23(2): 303-315." +reference_url: https://doi.org/10.1109/TEVC.2018.2855411 +optimum: "DTLZ3's front, whole; ideal point (0, 0, 0), nadir point (1, 1, 1); hypervolume 0.7971 (normalized objectives, reference point (1.1, 1.1, 1.1))" +languages: [rust, python] +order: 204 +family: DC-DTLZ +tab: DC2-DTLZ3 +--- + +# DC2-DTLZ3 with 3 objectives + +## The problem + +Li, Chen, Fu and Yao (2019) proposed C-TAEA, a two-archive algorithm for constrained problems, and, to test it, the DC-DTLZ problems: DTLZ1 and DTLZ3 with constraints on the decision variables, where Jain and Deb's C-DTLZ problems constrain the objectives. Type 1 (DC1) constrains the first variable and cuts the front into cones from the origin; type 2 (DC2) constrains the distance function g and leaves almost nothing but the front feasible, with local optima of the violation on the way; type 3 (DC3) does both, on every position variable. The paper's supplement defines them (section 1.2) with parameters a and b, and gives a = 3 and b = 0.5 for DC1 only; genoxide takes the others from the code of the authors' laboratory, EMOC, and the docs of `multi::problems::Dc1Dtlz1` say what else the supplement leaves open and how genoxide settles it. + +DC2-DTLZ3 with 3 objectives and 12 variables in [0, 1]: + +```text +minimize f₁ = (1 + g) cos(πx₁/2) cos(πx₂/2) + f₂ = (1 + g) cos(πx₁/2) sin(πx₂/2) + f₃ = (1 + g) sin(πx₁/2) + g = 100 (10 + Σᵢ₌₃¹² ((xᵢ − 0.5)² − cos(20π (xᵢ − 0.5)))) +subject to cos(aπg) ≥ b + e^(−g) ≥ b, a = 3, b = 0.5 +``` + +DTLZ3's front is the unit sphere's octant, reached with the distance variables at 0.5, where g = 0. With b = 0.5 the constraints ask g in [0, 1/9] or [5/9, ln 2], spheres of radius 1 to 10/9 and 14/9 to 1.69: feasible solutions lie within a thin shell of the front, and the front is DTLZ3's, whole. Its ideal point is (0, 0, 0) and its nadir point (1, 1, 1). The supplement gives no front; genoxide derives it (g = 0 is feasible under every constraint, and the position constraints don't involve g, so DTLZ3's front points are optimal wherever their position variables are feasible, and no other point is), and it agrees with the sampled front that EMOC ships. + +## What makes it hard + +The violation: b − cos(aπg) rises and falls with g, back to its least, b − 1, at g = 2/3, 4/3, …, where e^(−g) is the only constraint still broken. Constrained dominance ranks infeasible solutions by their violation alone, so a population that reaches one of these local minima of the violation stays there: to go on, it must climb out first. Almost nothing but the front is feasible, so it never finds a feasible solution to pull it down. + +## Representation + +A `Real` genome of 12 genes in [0, 1]. The problem is genoxide's `Dc2Dtlz3`, whose fitness is the three objectives and the total constraint violation, 0 when it is feasible; in Python, `gx.problems.Dc2Dtlz3()`. Solutions compare by constrained dominance: a feasible solution beats an infeasible one, of two infeasible ones the smaller violation wins, and of two feasible ones Pareto dominance decides. + +## Algorithm + +Two runs of NSGA-III, with a population of 92 for 2,000 generations each: + +- NSGA-III (Deb and Jain, 2014; Jain and Deb, 2014, IEEE Transactions on Evolutionary Computation 18(4): 577-601 and 602-622) with the settings that the C-TAEA paper gives its C-NSGA-III (supplement, tables 3 and 4): the 91 reference directions of Das and Dennis's method with 12 divisions and a population of 92, simulated binary crossover with η = 30 at a rate of 1, and polynomial mutation with η = 20 at a rate of 1/n per gene, for 2,000 generations (the supplement gives no budget for the DC-DTLZ problems; 2,000 on DTLZ3, whose local fronts take longer), with constrained dominance; +- the same on DTLZ3 itself, without the constraints, its final front then scored by DC2-DTLZ3: the front is DTLZ3's, and a population that converges to it is feasible. + +## Output + +A line per run: the size of its final front, how many of its solutions the problem finds feasible, their IGD+ and hypervolume, and the hypervolume as a share of that of a sample of the optimal front with at least as many points as the population. Then the hypervolumes of the whole front and of the sample. + +The indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 2,000 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest feasible solution of the found front, counting only the objectives in which the solution is worse. Smaller is better; a front of as many points as the population can't cover the 2,000 exactly, and the sample shows what it can. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the volume the feasible solutions dominate up to the reference point (1.1, 1.1, 1.1). Larger is better; the whole front's is computed from 3,000 of its points, and the sample's is what a front of that many points reaches. In Python, `run` evaluates the problem in Rust, so both versions print the same. + +The page plays the runs back side by side, each population as the problem scores it: its feasible non-dominated solutions, its infeasible ones (hollow, at most 100 a frame), and the optimal front, sampled. + +[The project page](https://tachsin.gr/projects/genoxide/examples/dc2-dtlz3-3obj) plays these runs back. + +## Good results + +The target: every solution feasible, and 99% of the hypervolume of the sample of 105 points of the front, about what a front of 92 solutions spread like it has. + +The run with constrained dominance ends with 92 solutions, none feasible, the least violation 0.5000: stuck at a local minimum of the violation. Without the constraints, it ends with 92 solutions, 92 feasible, IGD+ 0.0227, hypervolume 0.7437, 99.2% of the sample's. Over seeds 1 to 20, every run with constrained dominance ends with no feasible solution, and without the constraints IGD+ from 0.0223 to 0.0251 and 98.6% to 99.4% of the sample's hypervolume, 18 runs reaching the target. The C-TAEA paper's C-NSGA-III ends with a median IGD of 120 (its table 3), the same failure, and C-TAEA with 0.0550. Ignoring the constraints works here because they leave the front whole, as Tanabe and Oyama (2017) note of C1-DTLZ1 and C1-DTLZ3. diff --git a/examples/dc2_dtlz3_3obj/main.py b/examples/dc2_dtlz3_3obj/main.py new file mode 100644 index 00000000..394d9a1b --- /dev/null +++ b/examples/dc2_dtlz3_3obj/main.py @@ -0,0 +1,97 @@ +"""DC2-DTLZ3 with 3 objectives: minimize three objectives of DTLZ3 subject to two constraints on its +distance function, which leave almost nothing but the front feasible, with NSGA-III. + +From genoxide's problems.Dc2Dtlz3; run evaluates it in Rust. Runs NSGA-III with the settings of the +C-TAEA paper's C-NSGA-III, a population of 92 for 2,000 generations, twice: with constraint +dominance, and on DTLZ3 without the constraints. Prints each final front's size, how many of its +solutions are feasible, their IGD+ to 2,000 points of the optimal front and their hypervolume, with +the objectives normalized by the front's ideal and nadir points, as a share of that of a sample of +the front with at least as many points as the population; then the hypervolumes of the whole front +and of the sample. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/dc2_dtlz3_3obj/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points +REFERENCE = [1.1, 1.1, 1.1] + +problem = gx.problems.Dc2Dtlz3() +ideal, nadir = problem.ideal_point, problem.nadir_point +rate = 1 / problem.dimensions + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) +# the sample: at least as many points of the optimal front as the population has, about what a +# front of that many solutions can be +sample = normalized(problem.optimal_front(92)) + + +def run(name, algorithm, fitness, generations): + """Runs ``algorithm`` on ``fitness`` for ``generations``, and prints its final front's size, + how many of its solutions the problem finds feasible, their IGD+ to 2,000 points of the + optimal front and their hypervolume, with normalized objectives, as a share of the + sample's.""" + result = algorithm.run(fitness, generations=generations, on_generation=trace.front(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"{name}, {generations} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(2_000)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + percent = 100 * volume / gx.indicators.hypervolume(sample, REFERENCE) + print( + start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}, " + f"{percent:.1f}% of the sample's" + ) + + +# NSGA-III with the settings of the C-TAEA paper's C-NSGA-III, constraint dominance +algorithm = gx.Nsga3( + problem.genome, + objectives=problem.objectives, + reference_directions=gx.das_dennis(3, 12), + population_size=92, + crossover=gx.SimulatedBinaryCrossover(30), + crossover_rate=1.0, + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-III", algorithm, problem, 2_000) +# the same on DTLZ3, without the constraints, whose solutions the problem then scores +algorithm = gx.Nsga3( + problem.genome, + objectives=problem.objectives, + reference_directions=gx.das_dennis(3, 12), + population_size=92, + crossover=gx.SimulatedBinaryCrossover(30), + crossover_rate=1.0, + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-III, unconstrained", algorithm, gx.problems.Dtlz3(variables=problem.dimensions), 2_000) +whole = normalized(problem.optimal_front(3_000)) +print( + f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}; " + f"a sample of {len(sample)} of its points: " + f"{gx.indicators.hypervolume(sample, REFERENCE):.4f}" +) +trace.write() diff --git a/examples/dc2_dtlz3_3obj/main.rs b/examples/dc2_dtlz3_3obj/main.rs new file mode 100644 index 00000000..fc6352ab --- /dev/null +++ b/examples/dc2_dtlz3_3obj/main.rs @@ -0,0 +1,146 @@ +//! DC2-DTLZ3 with 3 objectives: minimize three objectives of DTLZ3 subject to two constraints on +//! its distance function, which leave almost nothing but the front feasible, with NSGA-III. +//! +//! From genoxide's `multi::problems::Dc2Dtlz3`. Runs NSGA-III with the settings of the C-TAEA +//! paper's C-NSGA-III, a population of 92 for 2,000 generations, twice: with constraint dominance, +//! and on DTLZ3 without the constraints. Prints each final front's size, how many of its solutions +//! are feasible, their IGD+ to 2,000 points of the optimal front and their hypervolume, with the +//! objectives normalized by the front's ideal and nadir points, as a share of that of a sample of +//! the front with at least as many points as the population; then the hypervolumes of the whole +//! front and of the sample. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example dc2_dtlz3_3obj +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{Dc2Dtlz3, Dtlz3, MultiProblem}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points +pub const REFERENCE: [f64; 3] = [1.1, 1.1, 1.1]; + +fn main() -> Result<()> { + let problem = Dc2Dtlz3::<3>::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the sample: at least as many points of the optimal front as the population has, about what + // a front of that many solutions can be + let sample = normalized(&problem, &problem.optimal_front(92).expect("known")); + let rate = 1.0 / problem.variables() as f64; + // NSGA-III with the settings of the C-TAEA paper's C-NSGA-III, constraint dominance + let algorithm = Nsga3::builder( + problem.representation(), + [Minimize; 3], + multi::das_dennis::<3>(12), + ) + .population_size(92) + .crossover(SimulatedBinaryCrossover::new(30.0)?) + .crossover_rate(1.0) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, "NSGA-III", algorithm, problem, 2_000, &sample, &mut trace, + )?; + // the same on DTLZ3, without the constraints, whose solutions the problem then scores + let algorithm = Nsga3::builder( + problem.representation(), + [Minimize; 3], + multi::das_dennis::<3>(12), + ) + .population_size(92) + .crossover(SimulatedBinaryCrossover::new(30.0)?) + .crossover_rate(1.0) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, + "NSGA-III, unconstrained", + algorithm, + Dtlz3::<3>::new(problem.variables()), + 2_000, + &sample, + &mut trace, + )?; + let whole = normalized(&problem, &problem.optimal_front(3_000).expect("known")); + println!( + "the whole front: hypervolume {:.4}; a sample of {} of its points: {:.4}", + hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + sample.len(), + hypervolume(&sample, &REFERENCE, &[Minimize; 3]) + ); + trace.write(&problem); + Ok(()) +} + +// runs `algorithm` on `fitness` for `generations`, and prints its final front's size, how many of +// its solutions the problem finds feasible, their IGD+ to 2,000 points of the optimal front and +// their hypervolume, with normalized objectives, as a share of the sample's +fn run( + problem: &Dc2Dtlz3<3>, + name: &'static str, + algorithm: A, + fitness: F, + generations: u64, + sample: &[[f64; 3]], + trace: &mut trace::Trace, +) -> Result<()> +where + A: MultiObjectiveAlgorithm<3, Genome = Reals>, + F: MultiFitnessFunction + Sync, +{ + let mut record = trace.front(name, *problem); + let outcome = MultiEngine::new(algorithm, fitness) + .stop_when(Stop::generations(generations)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 3], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "{name}, {generations} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(2_000).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 3]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 3]); + let percent = 100.0 * volume / hypervolume(sample, &REFERENCE, &[Minimize; 3]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the \ + sample's", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &Dc2Dtlz3<3>, points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 3]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/dc2_dtlz3_3obj/output.txt b/examples/dc2_dtlz3_3obj/output.txt new file mode 100644 index 00000000..1c728ab5 --- /dev/null +++ b/examples/dc2_dtlz3_3obj/output.txt @@ -0,0 +1,3 @@ +NSGA-III, 2000 generations: 92 solutions, none feasible, the least violation 0.5000 +NSGA-III, unconstrained, 2000 generations: 92 solutions, 92 feasible, IGD+ 0.0227, hypervolume 0.7437, 99.2% of the sample's +the whole front: hypervolume 0.7971; a sample of 105 of its points: 0.7494 diff --git a/examples/dc2_dtlz3_3obj/trace.json b/examples/dc2_dtlz3_3obj/trace.json new file mode 100644 index 00000000..d8339c83 --- /dev/null +++ b/examples/dc2_dtlz3_3obj/trace.json @@ -0,0 +1,42 @@ 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0.000228638],[0.896598,0.449231,2.15668e-05],[0.619455,0.778641,0.126853],[0.493157,0.866609,0.125614],[0.252648,0.792502,0.561568],[0.993597,0.0998055,0.0992849],[0.114169,0.924928,0.37136]],"infeasible":[]}]}} +]} diff --git a/examples/dc2_dtlz3_3obj/trace.py b/examples/dc2_dtlz3_3obj/trace.py new file mode 100644 index 00000000..0ad6ad50 --- /dev/null +++ b/examples/dc2_dtlz3_3obj/trace.py @@ -0,0 +1,215 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: each run's population as the problem scores it, its feasible +non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +front's hypervolume with normalized objectives, in at most 40 generations; a frame's +evaluations are the runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the most infeasible solutions a frame shows +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions and the front's hypervolume + self.series = {} + + def front(self, name): + """The callback for ``run`` that records the run ``name`` after each generation, its + population scored by the problem, which the run may not optimize as it is: None without a + trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + objectives, violations = self.problem.evaluate(progress.population) + front = non_dominated(objectives[violations == 0]) + infeasible = objectives[violations > 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + history.append((progress.evaluations, front.tolist(), infeasible.tolist(), volume)) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(40) + for generation in range(max(map(len, self.series.values()), default=0)): + at = [runs[min(generation, len(runs) - 1)] for runs in self.series.values()] + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at), + "best": None, + "median": None, + "series": {name: record[3] for name, record in zip(self.series, at)}, + "state": { + "panels": [{"front": record[1], "infeasible": record[2]} for record in at] + }, + } + ) + # the front, sampled, for every panel + panel = { + "objectives": ["f1", "f2", "f3"], + "true_front": self.problem.optimal_front(600).tolist(), + } + panels = [{"title": name, "problem": panel} for name in self.series] + whole = self.normalized(self.problem.optimal_front(3_000)) + settings = { + "format": 1, + "example": "dc2_dtlz3_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": list(self.series), + }, + } + write(self.path, settings, frames.to_list()) + + +def non_dominated(points): + """The points that no other point dominates, in their order.""" + keep = [ + not np.any(np.all(points <= point, axis=1) & np.any(points < point, axis=1)) + for point in points + ] + return points[np.array(keep, dtype=bool)] if len(points) else points + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/dc2_dtlz3_3obj/trace.rs b/examples/dc2_dtlz3_3obj/trace.rs new file mode 100644 index 00000000..f4afc85f --- /dev/null +++ b/examples/dc2_dtlz3_3obj/trace.rs @@ -0,0 +1,307 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: each run's population as the problem scores it, its feasible +//! non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +//! front's hypervolume with normalized objectives, in at most 40 generations; a frame's +//! evaluations are the runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::MultiSnapshot; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{Dc2Dtlz3, MultiProblem}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, and the +// hypervolume +type Record = (u64, Vec<[f64; 3]>, Vec<[f64; 3]>, f64); + +// the most infeasible solutions a frame shows +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation: its population scored by + // `problem`, which the run may not optimize as it is + pub fn front( + &mut self, + name: &'static str, + problem: Dc2Dtlz3<3>, + ) -> impl FnMut(&MultiSnapshot<'_, Reals, 3>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let population = snapshot.population().as_slice(); + let scores: Vec<([f64; 3], f64)> = population + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = + scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let front = non_dominated(&feasible); + let infeasible = scores.iter().filter(|s| s.1 > 0.0).map(|s| s.0); + let infeasible: Vec<[f64; 3]> = infeasible.take(SHOWN).collect(); + let volume = hypervolume(&normalized(&problem, &front), &REFERENCE, &[Minimize; 3]); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &Dc2Dtlz3<3>) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(40); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let (mut panels, mut volumes, mut evaluations) = (Vec::new(), Map::new(), 0); + for (name, history) in &self.series { + let (done, front, infeasible, volume) = &history[generation.min(history.len() - 1)]; + panels.push(json!({ "front": front, "infeasible": infeasible })); + volumes.insert(name.to_string(), json!(volume)); + evaluations += done; + } + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "series": volumes, + "state": { "panels": panels }, + })); + } + // the front, sampled, for every panel + let panel = json!({ + "objectives": ["f1", "f2", "f3"], + "true_front": problem.optimal_front(600).expect("known"), + }); + let panels: Vec = self + .series + .iter() + .map(|(name, _)| json!({ "title": name, "problem": panel })) + .collect(); + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(3_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "dc2_dtlz3_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": names, + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the points that no other point dominates, in their order +fn non_dominated(points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let dominates = + |p: &[f64; 3], q: &[f64; 3]| (0..3).all(|j| p[j] <= q[j]) && (0..3).any(|j| p[j] < q[j]); + let kept = points + .iter() + .filter(|q| !points.iter().any(|p| dominates(p, q))); + kept.copied().collect() +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/dc3_dtlz1_3obj/README.md b/examples/dc3_dtlz1_3obj/README.md new file mode 100644 index 00000000..d5f5a68a --- /dev/null +++ b/examples/dc3_dtlz1_3obj/README.md @@ -0,0 +1,63 @@ +--- +title: DC3-DTLZ1 with 3 objectives +category: multi-objective +summary: Minimize three objectives of DTLZ1 subject to constraints on its position variables and its distance function, whose Pareto front is four patches, with NSGA-III; constrained dominance stalls in a band of the distance function, and dropping that constraint reaches the front. +reference: "Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for constrained multiobjective optimization. IEEE Transactions on Evolutionary Computation 23(2): 303-315." +reference_url: https://doi.org/10.1109/TEVC.2018.2855411 +optimum: "four patches of DTLZ1's front; ideal point (0, 0, 1/9), nadir point (49/162, 7/18, 1/2) = (0.3025, 0.3889, 0.5); hypervolume 0.9010 (normalized objectives, reference point (1.1, 1.1, 1.1))" +languages: [rust, python] +order: 205 +family: DC-DTLZ +tab: DC3-DTLZ1 +--- + +# DC3-DTLZ1 with 3 objectives + +## The problem + +Li, Chen, Fu and Yao (2019) proposed C-TAEA, a two-archive algorithm for constrained problems, and, to test it, the DC-DTLZ problems: DTLZ1 and DTLZ3 with constraints on the decision variables, where Jain and Deb's C-DTLZ problems constrain the objectives. Type 1 (DC1) constrains the first variable and cuts the front into cones from the origin; type 2 (DC2) constrains the distance function g and leaves almost nothing but the front feasible, with local optima of the violation on the way; type 3 (DC3) does both, on every position variable. The paper's supplement defines them (section 1.2) with parameters a and b, and gives a = 3 and b = 0.5 for DC1 only; genoxide takes the others from the code of the authors' laboratory, EMOC, and the docs of `multi::problems::Dc1Dtlz1` say what else the supplement leaves open and how genoxide settles it. + +DC3-DTLZ1 with 3 objectives and 7 variables in [0, 1]: + +```text +minimize f₁ = ½ x₁ x₂ (1 + g) + f₂ = ½ x₁ (1 − x₂) (1 + g) + f₃ = ½ (1 − x₁) (1 + g) + g = 100 (5 + Σᵢ₌₃⁷ ((xᵢ − 0.5)² − cos(20π (xᵢ − 0.5)))) +subject to cos(aπx₁) ≥ b + cos(aπx₂) ≥ b + cos(aπg) ≥ b, a = 3, b = 0.5 +``` + +DTLZ1's front is the triangle where the objectives sum to 1/2, reached with the distance variables at 0.5, where g = 0. Each position variable must be in [0, 1/9] or [5/9, 7/9], and g in [0, 1/9], [5/9, 7/9], [11/9, 13/9], …: the front is the parts of DTLZ1's where both position variables are, four patches. (The supplement writes the position constraints for j = 1, …, m, which would include the first distance variable, 0.5 on the front, and make it all infeasible; genoxide constrains the m − 1 position variables, as EMOC does and the supplement's table 2 confirms.) Its ideal point is (0, 0, 1/9) and its nadir point (49/162, 7/18, 1/2) = (0.3025, 0.3889, 0.5). The supplement gives no front; genoxide derives it (g = 0 is feasible under every constraint, and the position constraints don't involve g, so DTLZ1's front points are optimal wherever their position variables are feasible, and no other point is), and it agrees with the sampled front that EMOC ships. + +## What makes it hard + +The constraint on g: it's feasible in bands, g in [0, 1/9], [5/9, 7/9], [11/9, 13/9], …, and with DTLZ's g, 100 times a sum, a population finds a band far from the front, feasible, and stays in it, as constrained dominance accepts no step through the infeasible gap below. Then the patches, a ninth of the front, which the population must spread over. + +## Representation + +A `Real` genome of 7 genes in [0, 1]. The problem is genoxide's `Dc3Dtlz1`, whose fitness is the three objectives and the total constraint violation, 0 when it is feasible; in Python, `gx.problems.Dc3Dtlz1()`. Solutions compare by constrained dominance: a feasible solution beats an infeasible one, of two infeasible ones the smaller violation wins, and of two feasible ones Pareto dominance decides. + +## Algorithm + +Two runs of NSGA-III, with a population of 92 for 1,000 generations each: + +- NSGA-III (Deb and Jain, 2014; Jain and Deb, 2014, IEEE Transactions on Evolutionary Computation 18(4): 577-601 and 602-622) with the settings that the C-TAEA paper gives its C-NSGA-III (supplement, tables 3 and 4): the 91 reference directions of Das and Dennis's method with 12 divisions and a population of 92, simulated binary crossover with η = 30 at a rate of 1, and polynomial mutation with η = 20 at a rate of 1/n per gene, for 1,000 generations (the supplement gives no budget for the DC-DTLZ problems; 2,000 on DTLZ3, whose local fronts take longer), with constrained dominance; +- the same without the constraint on g, the position constraints only, its final front then scored by DC3-DTLZ1: the constraint on g holds at g = 0, so a population that converges to the front is feasible. + +## Output + +A line per run: the size of its final front, how many of its solutions the problem finds feasible, their IGD+ and hypervolume, and the hypervolume as a share of that of a sample of the optimal front with at least as many points as the population. Then the hypervolumes of the whole front and of the sample. + +The indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 2,000 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest feasible solution of the found front, counting only the objectives in which the solution is worse. Smaller is better; a front of as many points as the population can't cover the 2,000 exactly, and the sample shows what it can. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the volume the feasible solutions dominate up to the reference point (1.1, 1.1, 1.1). Larger is better; the whole front's is computed from 3,000 of its points, and the sample's is what a front of that many points reaches. In Python, `run` evaluates the problem in Rust, so both versions print the same. + +The page plays the runs back side by side, each population as the problem scores it: its feasible non-dominated solutions, its infeasible ones (hollow, at most 100 a frame), and the optimal front, sampled. + +[The project page](https://tachsin.gr/projects/genoxide/examples/dc3-dtlz1-3obj) plays these runs back. + +## Good results + +The target: every solution feasible, and 99% of the hypervolume of the sample of 93 points of the front, about what a front of 92 solutions spread like it has. + +The run with constrained dominance ends with 92 solutions, 92 feasible, IGD+ 0.4606, hypervolume 0.1355, 15.6% of the sample's: in a band of g away from the front. Without the constraint on g, it ends with 92 solutions, 92 feasible, IGD+ 0.0212, hypervolume 0.8726, 100.6% of the sample's. Over seeds 1 to 20, constrained dominance ends with IGD+ from 0.0202 to 1.0190, a median of 0.4597, and without the constraint on g with IGD+ from 0.0197 to 0.2537 and 76.3% to 101.9% of the sample's hypervolume, 18 runs reaching the target. The C-TAEA paper's C-NSGA-III ends with a median IGD of 9.75 (its table 3), the same failure, and C-TAEA with 0.0503. diff --git a/examples/dc3_dtlz1_3obj/main.py b/examples/dc3_dtlz1_3obj/main.py new file mode 100644 index 00000000..598430f2 --- /dev/null +++ b/examples/dc3_dtlz1_3obj/main.py @@ -0,0 +1,106 @@ +"""DC3-DTLZ1 with 3 objectives: minimize three objectives of DTLZ1 subject to constraints on its +position variables and its distance function, whose front is patches of DTLZ's, with NSGA-III. + +From genoxide's problems.Dc3Dtlz1; run evaluates it in Rust. Runs NSGA-III with the settings of the +C-TAEA paper's C-NSGA-III, a population of 92 for 1,000 generations, twice: with constraint +dominance, and without the constraint on the distance function. Prints each final front's size, how +many of its solutions are feasible, their IGD+ to 2,000 points of the optimal front and their +hypervolume, with the objectives normalized by the front's ideal and nadir points, as a share of +that of a sample of the front with at least as many points as the population; then the hypervolumes +of the whole front and of the sample. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/dc3_dtlz1_3obj/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points +REFERENCE = [1.1, 1.1, 1.1] + +problem = gx.problems.Dc3Dtlz1() +ideal, nadir = problem.ideal_point, problem.nadir_point +rate = 1 / problem.dimensions + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) +# the sample: at least as many points of the optimal front as the population has, about what a +# front of that many solutions can be +sample = normalized(problem.optimal_front(92)) + + +def without_g(x): + """The problem without its last constraint, the one on the distance function g: its + objectives, and the violation of the constraints on the position variables alone.""" + objectives, _ = problem(x) + constraints = problem.constraints(x) + return objectives, sum(max(float(g), 0.0) for g in constraints[:2]) + + + +def run(name, algorithm, fitness, generations): + """Runs ``algorithm`` on ``fitness`` for ``generations``, and prints its final front's size, + how many of its solutions the problem finds feasible, their IGD+ to 2,000 points of the + optimal front and their hypervolume, with normalized objectives, as a share of the + sample's.""" + result = algorithm.run(fitness, generations=generations, on_generation=trace.front(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"{name}, {generations} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(2_000)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + percent = 100 * volume / gx.indicators.hypervolume(sample, REFERENCE) + print( + start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}, " + f"{percent:.1f}% of the sample's" + ) + + +# NSGA-III with the settings of the C-TAEA paper's C-NSGA-III, constraint dominance +algorithm = gx.Nsga3( + problem.genome, + objectives=problem.objectives, + reference_directions=gx.das_dennis(3, 12), + population_size=92, + crossover=gx.SimulatedBinaryCrossover(30), + crossover_rate=1.0, + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-III", algorithm, problem, 1_000) +# the same without the constraint on g, whose solutions the problem then scores +algorithm = gx.Nsga3( + problem.genome, + objectives=problem.objectives, + reference_directions=gx.das_dennis(3, 12), + population_size=92, + crossover=gx.SimulatedBinaryCrossover(30), + crossover_rate=1.0, + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-III, no constraint on g", algorithm, without_g, 1_000) +whole = normalized(problem.optimal_front(3_000)) +print( + f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}; " + f"a sample of {len(sample)} of its points: " + f"{gx.indicators.hypervolume(sample, REFERENCE):.4f}" +) +trace.write() diff --git a/examples/dc3_dtlz1_3obj/main.rs b/examples/dc3_dtlz1_3obj/main.rs new file mode 100644 index 00000000..3201e5c2 --- /dev/null +++ b/examples/dc3_dtlz1_3obj/main.rs @@ -0,0 +1,156 @@ +//! DC3-DTLZ1 with 3 objectives: minimize three objectives of DTLZ1 subject to constraints on its +//! position variables and its distance function, whose front is patches of DTLZ's, with NSGA-III. +//! +//! From genoxide's `multi::problems::Dc3Dtlz1`. Runs NSGA-III with the settings of the C-TAEA +//! paper's C-NSGA-III, a population of 92 for 1,000 generations, twice: with constraint dominance, +//! and without the constraint on the distance function. Prints each final front's size, how many of +//! its solutions are feasible, their IGD+ to 2,000 points of the optimal front and their +//! hypervolume, with the objectives normalized by the front's ideal and nadir points, as a share of +//! that of a sample of the front with at least as many points as the population; then the +//! hypervolumes of the whole front and of the sample. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example dc3_dtlz1_3obj +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{Dc3Dtlz1, MultiProblem}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points +pub const REFERENCE: [f64; 3] = [1.1, 1.1, 1.1]; + +fn main() -> Result<()> { + let problem = Dc3Dtlz1::<3>::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the sample: at least as many points of the optimal front as the population has, about what + // a front of that many solutions can be + let sample = normalized(&problem, &problem.optimal_front(92).expect("known")); + let rate = 1.0 / problem.variables() as f64; + // NSGA-III with the settings of the C-TAEA paper's C-NSGA-III, constraint dominance + let algorithm = Nsga3::builder( + problem.representation(), + [Minimize; 3], + multi::das_dennis::<3>(12), + ) + .population_size(92) + .crossover(SimulatedBinaryCrossover::new(30.0)?) + .crossover_rate(1.0) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, "NSGA-III", algorithm, problem, 1_000, &sample, &mut trace, + )?; + // the same without the constraint on g, whose solutions the problem then scores + let algorithm = Nsga3::builder( + problem.representation(), + [Minimize; 3], + multi::das_dennis::<3>(12), + ) + .population_size(92) + .crossover(SimulatedBinaryCrossover::new(30.0)?) + .crossover_rate(1.0) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, + "NSGA-III, no constraint on g", + algorithm, + |x: &Reals| without_g(&problem, x), + 1_000, + &sample, + &mut trace, + )?; + let whole = normalized(&problem, &problem.optimal_front(3_000).expect("known")); + println!( + "the whole front: hypervolume {:.4}; a sample of {} of its points: {:.4}", + hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + sample.len(), + hypervolume(&sample, &REFERENCE, &[Minimize; 3]) + ); + trace.write(&problem); + Ok(()) +} + +// the problem without its last constraint, the one on the distance function g: its objectives, +// and the violation of the constraints on the position variables alone +fn without_g(problem: &Dc3Dtlz1<3>, x: &Reals) -> ([f64; 3], f64) { + let (objectives, _) = problem.evaluate(x); + let constraints = problem.constraints(x); + let positions = &constraints.inequalities()[..2]; + let violation = positions.iter().map(|g| g.max(0.0)).sum(); + (objectives, violation) +} + +// runs `algorithm` on `fitness` for `generations`, and prints its final front's size, how many of +// its solutions the problem finds feasible, their IGD+ to 2,000 points of the optimal front and +// their hypervolume, with normalized objectives, as a share of the sample's +fn run( + problem: &Dc3Dtlz1<3>, + name: &'static str, + algorithm: A, + fitness: F, + generations: u64, + sample: &[[f64; 3]], + trace: &mut trace::Trace, +) -> Result<()> +where + A: MultiObjectiveAlgorithm<3, Genome = Reals>, + F: MultiFitnessFunction + Sync, +{ + let mut record = trace.front(name, *problem); + let outcome = MultiEngine::new(algorithm, fitness) + .stop_when(Stop::generations(generations)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 3], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "{name}, {generations} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(2_000).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 3]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 3]); + let percent = 100.0 * volume / hypervolume(sample, &REFERENCE, &[Minimize; 3]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the \ + sample's", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &Dc3Dtlz1<3>, points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 3]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/dc3_dtlz1_3obj/output.txt b/examples/dc3_dtlz1_3obj/output.txt new file mode 100644 index 00000000..da355604 --- /dev/null +++ b/examples/dc3_dtlz1_3obj/output.txt @@ -0,0 +1,3 @@ +NSGA-III, 1000 generations: 92 solutions, 92 feasible, IGD+ 0.4606, hypervolume 0.1355, 15.6% of the sample's +NSGA-III, no constraint on g, 1000 generations: 92 solutions, 92 feasible, IGD+ 0.0212, hypervolume 0.8726, 100.6% of the sample's +the whole front: hypervolume 0.9010; a sample of 93 of its points: 0.8671 diff --git a/examples/dc3_dtlz1_3obj/trace.json b/examples/dc3_dtlz1_3obj/trace.json new file mode 100644 index 00000000..8091610c --- /dev/null +++ b/examples/dc3_dtlz1_3obj/trace.json @@ -0,0 +1,42 @@ 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+]} diff --git a/examples/dc3_dtlz1_3obj/trace.py b/examples/dc3_dtlz1_3obj/trace.py new file mode 100644 index 00000000..1b23269a --- /dev/null +++ b/examples/dc3_dtlz1_3obj/trace.py @@ -0,0 +1,215 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: each run's population as the problem scores it, its feasible +non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +front's hypervolume with normalized objectives, in at most 40 generations; a frame's +evaluations are the runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the most infeasible solutions a frame shows +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions and the front's hypervolume + self.series = {} + + def front(self, name): + """The callback for ``run`` that records the run ``name`` after each generation, its + population scored by the problem, which the run may not optimize as it is: None without a + trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + objectives, violations = self.problem.evaluate(progress.population) + front = non_dominated(objectives[violations == 0]) + infeasible = objectives[violations > 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + history.append((progress.evaluations, front.tolist(), infeasible.tolist(), volume)) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(40) + for generation in range(max(map(len, self.series.values()), default=0)): + at = [runs[min(generation, len(runs) - 1)] for runs in self.series.values()] + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at), + "best": None, + "median": None, + "series": {name: record[3] for name, record in zip(self.series, at)}, + "state": { + "panels": [{"front": record[1], "infeasible": record[2]} for record in at] + }, + } + ) + # the front, sampled, for every panel + panel = { + "objectives": ["f1", "f2", "f3"], + "true_front": self.problem.optimal_front(600).tolist(), + } + panels = [{"title": name, "problem": panel} for name in self.series] + whole = self.normalized(self.problem.optimal_front(3_000)) + settings = { + "format": 1, + "example": "dc3_dtlz1_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": list(self.series), + }, + } + write(self.path, settings, frames.to_list()) + + +def non_dominated(points): + """The points that no other point dominates, in their order.""" + keep = [ + not np.any(np.all(points <= point, axis=1) & np.any(points < point, axis=1)) + for point in points + ] + return points[np.array(keep, dtype=bool)] if len(points) else points + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/dc3_dtlz1_3obj/trace.rs b/examples/dc3_dtlz1_3obj/trace.rs new file mode 100644 index 00000000..f1b5ddd7 --- /dev/null +++ b/examples/dc3_dtlz1_3obj/trace.rs @@ -0,0 +1,307 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: each run's population as the problem scores it, its feasible +//! non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +//! front's hypervolume with normalized objectives, in at most 40 generations; a frame's +//! evaluations are the runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::MultiSnapshot; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{Dc3Dtlz1, MultiProblem}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, and the +// hypervolume +type Record = (u64, Vec<[f64; 3]>, Vec<[f64; 3]>, f64); + +// the most infeasible solutions a frame shows +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation: its population scored by + // `problem`, which the run may not optimize as it is + pub fn front( + &mut self, + name: &'static str, + problem: Dc3Dtlz1<3>, + ) -> impl FnMut(&MultiSnapshot<'_, Reals, 3>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let population = snapshot.population().as_slice(); + let scores: Vec<([f64; 3], f64)> = population + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = + scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let front = non_dominated(&feasible); + let infeasible = scores.iter().filter(|s| s.1 > 0.0).map(|s| s.0); + let infeasible: Vec<[f64; 3]> = infeasible.take(SHOWN).collect(); + let volume = hypervolume(&normalized(&problem, &front), &REFERENCE, &[Minimize; 3]); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &Dc3Dtlz1<3>) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(40); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let (mut panels, mut volumes, mut evaluations) = (Vec::new(), Map::new(), 0); + for (name, history) in &self.series { + let (done, front, infeasible, volume) = &history[generation.min(history.len() - 1)]; + panels.push(json!({ "front": front, "infeasible": infeasible })); + volumes.insert(name.to_string(), json!(volume)); + evaluations += done; + } + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "series": volumes, + "state": { "panels": panels }, + })); + } + // the front, sampled, for every panel + let panel = json!({ + "objectives": ["f1", "f2", "f3"], + "true_front": problem.optimal_front(600).expect("known"), + }); + let panels: Vec = self + .series + .iter() + .map(|(name, _)| json!({ "title": name, "problem": panel })) + .collect(); + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(3_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "dc3_dtlz1_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": names, + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the points that no other point dominates, in their order +fn non_dominated(points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let dominates = + |p: &[f64; 3], q: &[f64; 3]| (0..3).all(|j| p[j] <= q[j]) && (0..3).any(|j| p[j] < q[j]); + let kept = points + .iter() + .filter(|q| !points.iter().any(|p| dominates(p, q))); + kept.copied().collect() +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/dc3_dtlz3_3obj/README.md b/examples/dc3_dtlz3_3obj/README.md new file mode 100644 index 00000000..d92a718c --- /dev/null +++ b/examples/dc3_dtlz3_3obj/README.md @@ -0,0 +1,63 @@ +--- +title: DC3-DTLZ3 with 3 objectives +category: multi-objective +summary: Minimize three objectives of DTLZ3 subject to constraints on its position variables and its distance function, whose Pareto front is four patches, with NSGA-III; constrained dominance stalls in a band of the distance function, and dropping that constraint reaches the front. +reference: "Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for constrained multiobjective optimization. IEEE Transactions on Evolutionary Computation 23(2): 303-315." +reference_url: https://doi.org/10.1109/TEVC.2018.2855411 +optimum: "four patches of DTLZ3's front; ideal point (cos²(7π/18), 0, 0) = (0.117, 0, 0), nadir point (1, sin(7π/18), sin(7π/18)) = (1, 0.9397, 0.9397); hypervolume 0.5488 (normalized objectives, reference point (1.1, 1.1, 1.1))" +languages: [rust, python] +order: 206 +family: DC-DTLZ +tab: DC3-DTLZ3 +--- + +# DC3-DTLZ3 with 3 objectives + +## The problem + +Li, Chen, Fu and Yao (2019) proposed C-TAEA, a two-archive algorithm for constrained problems, and, to test it, the DC-DTLZ problems: DTLZ1 and DTLZ3 with constraints on the decision variables, where Jain and Deb's C-DTLZ problems constrain the objectives. Type 1 (DC1) constrains the first variable and cuts the front into cones from the origin; type 2 (DC2) constrains the distance function g and leaves almost nothing but the front feasible, with local optima of the violation on the way; type 3 (DC3) does both, on every position variable. The paper's supplement defines them (section 1.2) with parameters a and b, and gives a = 3 and b = 0.5 for DC1 only; genoxide takes the others from the code of the authors' laboratory, EMOC, and the docs of `multi::problems::Dc1Dtlz1` say what else the supplement leaves open and how genoxide settles it. + +DC3-DTLZ3 with 3 objectives and 12 variables in [0, 1]: + +```text +minimize f₁ = (1 + g) cos(πx₁/2) cos(πx₂/2) + f₂ = (1 + g) cos(πx₁/2) sin(πx₂/2) + f₃ = (1 + g) sin(πx₁/2) + g = 100 (10 + Σᵢ₌₃¹² ((xᵢ − 0.5)² − cos(20π (xᵢ − 0.5)))) +subject to cos(aπx₁) ≥ b + cos(aπx₂) ≥ b + cos(aπg) ≥ b, a = 3, b = 0.5 +``` + +DTLZ3's front is the unit sphere's octant, reached with the distance variables at 0.5, where g = 0. Each position variable must be in [0, 1/9] or [5/9, 7/9], and g in [0, 1/9], [5/9, 7/9], [11/9, 13/9], …: the front is the parts of DTLZ3's where both position variables are, four patches. (The supplement writes the position constraints for j = 1, …, m, which would include the first distance variable, 0.5 on the front, and make it all infeasible; genoxide constrains the m − 1 position variables, as EMOC does and the supplement's table 2 confirms.) Its ideal point is (cos²(7π/18), 0, 0) = (0.117, 0, 0) and its nadir point (1, sin(7π/18), sin(7π/18)) = (1, 0.9397, 0.9397). The supplement gives no front; genoxide derives it (g = 0 is feasible under every constraint, and the position constraints don't involve g, so DTLZ3's front points are optimal wherever their position variables are feasible, and no other point is), and it agrees with the sampled front that EMOC ships. + +## What makes it hard + +The constraint on g: it's feasible in bands, g in [0, 1/9], [5/9, 7/9], [11/9, 13/9], …, and with DTLZ's g, 100 times a sum, a population finds a band far from the front, feasible, and stays in it, as constrained dominance accepts no step through the infeasible gap below. Then the patches, a ninth of the front, which the population must spread over. + +## Representation + +A `Real` genome of 12 genes in [0, 1]. The problem is genoxide's `Dc3Dtlz3`, whose fitness is the three objectives and the total constraint violation, 0 when it is feasible; in Python, `gx.problems.Dc3Dtlz3()`. Solutions compare by constrained dominance: a feasible solution beats an infeasible one, of two infeasible ones the smaller violation wins, and of two feasible ones Pareto dominance decides. + +## Algorithm + +Two runs of NSGA-III, with a population of 92 for 2,000 generations each: + +- NSGA-III (Deb and Jain, 2014; Jain and Deb, 2014, IEEE Transactions on Evolutionary Computation 18(4): 577-601 and 602-622) with the settings that the C-TAEA paper gives its C-NSGA-III (supplement, tables 3 and 4): the 91 reference directions of Das and Dennis's method with 12 divisions and a population of 92, simulated binary crossover with η = 30 at a rate of 1, and polynomial mutation with η = 20 at a rate of 1/n per gene, for 2,000 generations (the supplement gives no budget for the DC-DTLZ problems; 2,000 on DTLZ3, whose local fronts take longer), with constrained dominance; +- the same without the constraint on g, the position constraints only, its final front then scored by DC3-DTLZ3: the constraint on g holds at g = 0, so a population that converges to the front is feasible. + +## Output + +A line per run: the size of its final front, how many of its solutions the problem finds feasible, their IGD+ and hypervolume, and the hypervolume as a share of that of a sample of the optimal front with at least as many points as the population. Then the hypervolumes of the whole front and of the sample. + +The indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 2,000 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest feasible solution of the found front, counting only the objectives in which the solution is worse. Smaller is better; a front of as many points as the population can't cover the 2,000 exactly, and the sample shows what it can. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the volume the feasible solutions dominate up to the reference point (1.1, 1.1, 1.1). Larger is better; the whole front's is computed from 3,000 of its points, and the sample's is what a front of that many points reaches. In Python, `run` evaluates the problem in Rust, so both versions print the same. + +The page plays the runs back side by side, each population as the problem scores it: its feasible non-dominated solutions, its infeasible ones (hollow, at most 100 a frame), and the optimal front, sampled. + +[The project page](https://tachsin.gr/projects/genoxide/examples/dc3-dtlz3-3obj) plays these runs back. + +## Good results + +The target: every solution feasible, and 99% of the hypervolume of the sample of 94 points of the front, about what a front of 92 solutions spread like it has. + +The run with constrained dominance ends with 92 solutions, 92 feasible, IGD+ 0.6081, hypervolume 0.0000, 0.0% of the sample's: in a band of g away from the front. Without the constraint on g, it ends with 92 solutions, 92 feasible, IGD+ 0.0123, hypervolume 0.5303, 101.4% of the sample's. Over seeds 1 to 20, constrained dominance ends with IGD+ from 0.0136 to 3.5145, a median of 0.6134, and without the constraint on g with IGD+ from 0.0113 to 0.0148 and 99.2% to 101.5% of the sample's hypervolume, every run reaching the target. The C-TAEA paper's C-NSGA-III ends with a median IGD of 33.3 (its table 3), the same failure, and C-TAEA with 0.125. diff --git a/examples/dc3_dtlz3_3obj/main.py b/examples/dc3_dtlz3_3obj/main.py new file mode 100644 index 00000000..af065681 --- /dev/null +++ b/examples/dc3_dtlz3_3obj/main.py @@ -0,0 +1,106 @@ +"""DC3-DTLZ3 with 3 objectives: minimize three objectives of DTLZ3 subject to constraints on its +position variables and its distance function, whose front is patches of DTLZ's, with NSGA-III. + +From genoxide's problems.Dc3Dtlz3; run evaluates it in Rust. Runs NSGA-III with the settings of the +C-TAEA paper's C-NSGA-III, a population of 92 for 2,000 generations, twice: with constraint +dominance, and without the constraint on the distance function. Prints each final front's size, how +many of its solutions are feasible, their IGD+ to 2,000 points of the optimal front and their +hypervolume, with the objectives normalized by the front's ideal and nadir points, as a share of +that of a sample of the front with at least as many points as the population; then the hypervolumes +of the whole front and of the sample. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/dc3_dtlz3_3obj/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points +REFERENCE = [1.1, 1.1, 1.1] + +problem = gx.problems.Dc3Dtlz3() +ideal, nadir = problem.ideal_point, problem.nadir_point +rate = 1 / problem.dimensions + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) +# the sample: at least as many points of the optimal front as the population has, about what a +# front of that many solutions can be +sample = normalized(problem.optimal_front(92)) + + +def without_g(x): + """The problem without its last constraint, the one on the distance function g: its + objectives, and the violation of the constraints on the position variables alone.""" + objectives, _ = problem(x) + constraints = problem.constraints(x) + return objectives, sum(max(float(g), 0.0) for g in constraints[:2]) + + + +def run(name, algorithm, fitness, generations): + """Runs ``algorithm`` on ``fitness`` for ``generations``, and prints its final front's size, + how many of its solutions the problem finds feasible, their IGD+ to 2,000 points of the + optimal front and their hypervolume, with normalized objectives, as a share of the + sample's.""" + result = algorithm.run(fitness, generations=generations, on_generation=trace.front(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"{name}, {generations} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(2_000)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + percent = 100 * volume / gx.indicators.hypervolume(sample, REFERENCE) + print( + start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}, " + f"{percent:.1f}% of the sample's" + ) + + +# NSGA-III with the settings of the C-TAEA paper's C-NSGA-III, constraint dominance +algorithm = gx.Nsga3( + problem.genome, + objectives=problem.objectives, + reference_directions=gx.das_dennis(3, 12), + population_size=92, + crossover=gx.SimulatedBinaryCrossover(30), + crossover_rate=1.0, + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-III", algorithm, problem, 2_000) +# the same without the constraint on g, whose solutions the problem then scores +algorithm = gx.Nsga3( + problem.genome, + objectives=problem.objectives, + reference_directions=gx.das_dennis(3, 12), + population_size=92, + crossover=gx.SimulatedBinaryCrossover(30), + crossover_rate=1.0, + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-III, no constraint on g", algorithm, without_g, 2_000) +whole = normalized(problem.optimal_front(3_000)) +print( + f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}; " + f"a sample of {len(sample)} of its points: " + f"{gx.indicators.hypervolume(sample, REFERENCE):.4f}" +) +trace.write() diff --git a/examples/dc3_dtlz3_3obj/main.rs b/examples/dc3_dtlz3_3obj/main.rs new file mode 100644 index 00000000..07571f15 --- /dev/null +++ b/examples/dc3_dtlz3_3obj/main.rs @@ -0,0 +1,156 @@ +//! DC3-DTLZ3 with 3 objectives: minimize three objectives of DTLZ3 subject to constraints on its +//! position variables and its distance function, whose front is patches of DTLZ's, with NSGA-III. +//! +//! From genoxide's `multi::problems::Dc3Dtlz3`. Runs NSGA-III with the settings of the C-TAEA +//! paper's C-NSGA-III, a population of 92 for 2,000 generations, twice: with constraint dominance, +//! and without the constraint on the distance function. Prints each final front's size, how many of +//! its solutions are feasible, their IGD+ to 2,000 points of the optimal front and their +//! hypervolume, with the objectives normalized by the front's ideal and nadir points, as a share of +//! that of a sample of the front with at least as many points as the population; then the +//! hypervolumes of the whole front and of the sample. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example dc3_dtlz3_3obj +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{Dc3Dtlz3, MultiProblem}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points +pub const REFERENCE: [f64; 3] = [1.1, 1.1, 1.1]; + +fn main() -> Result<()> { + let problem = Dc3Dtlz3::<3>::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the sample: at least as many points of the optimal front as the population has, about what + // a front of that many solutions can be + let sample = normalized(&problem, &problem.optimal_front(92).expect("known")); + let rate = 1.0 / problem.variables() as f64; + // NSGA-III with the settings of the C-TAEA paper's C-NSGA-III, constraint dominance + let algorithm = Nsga3::builder( + problem.representation(), + [Minimize; 3], + multi::das_dennis::<3>(12), + ) + .population_size(92) + .crossover(SimulatedBinaryCrossover::new(30.0)?) + .crossover_rate(1.0) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, "NSGA-III", algorithm, problem, 2_000, &sample, &mut trace, + )?; + // the same without the constraint on g, whose solutions the problem then scores + let algorithm = Nsga3::builder( + problem.representation(), + [Minimize; 3], + multi::das_dennis::<3>(12), + ) + .population_size(92) + .crossover(SimulatedBinaryCrossover::new(30.0)?) + .crossover_rate(1.0) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, + "NSGA-III, no constraint on g", + algorithm, + |x: &Reals| without_g(&problem, x), + 2_000, + &sample, + &mut trace, + )?; + let whole = normalized(&problem, &problem.optimal_front(3_000).expect("known")); + println!( + "the whole front: hypervolume {:.4}; a sample of {} of its points: {:.4}", + hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + sample.len(), + hypervolume(&sample, &REFERENCE, &[Minimize; 3]) + ); + trace.write(&problem); + Ok(()) +} + +// the problem without its last constraint, the one on the distance function g: its objectives, +// and the violation of the constraints on the position variables alone +fn without_g(problem: &Dc3Dtlz3<3>, x: &Reals) -> ([f64; 3], f64) { + let (objectives, _) = problem.evaluate(x); + let constraints = problem.constraints(x); + let positions = &constraints.inequalities()[..2]; + let violation = positions.iter().map(|g| g.max(0.0)).sum(); + (objectives, violation) +} + +// runs `algorithm` on `fitness` for `generations`, and prints its final front's size, how many of +// its solutions the problem finds feasible, their IGD+ to 2,000 points of the optimal front and +// their hypervolume, with normalized objectives, as a share of the sample's +fn run( + problem: &Dc3Dtlz3<3>, + name: &'static str, + algorithm: A, + fitness: F, + generations: u64, + sample: &[[f64; 3]], + trace: &mut trace::Trace, +) -> Result<()> +where + A: MultiObjectiveAlgorithm<3, Genome = Reals>, + F: MultiFitnessFunction + Sync, +{ + let mut record = trace.front(name, *problem); + let outcome = MultiEngine::new(algorithm, fitness) + .stop_when(Stop::generations(generations)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 3], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "{name}, {generations} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(2_000).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 3]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 3]); + let percent = 100.0 * volume / hypervolume(sample, &REFERENCE, &[Minimize; 3]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the \ + sample's", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &Dc3Dtlz3<3>, points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 3]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/dc3_dtlz3_3obj/output.txt b/examples/dc3_dtlz3_3obj/output.txt new file mode 100644 index 00000000..e851b6b8 --- /dev/null +++ b/examples/dc3_dtlz3_3obj/output.txt @@ -0,0 +1,3 @@ +NSGA-III, 2000 generations: 92 solutions, 92 feasible, IGD+ 0.6081, hypervolume 0.0000, 0.0% of the sample's +NSGA-III, no constraint on g, 2000 generations: 92 solutions, 92 feasible, IGD+ 0.0123, hypervolume 0.5303, 101.4% of the sample's +the whole front: hypervolume 0.5488; a sample of 94 of its points: 0.5232 diff --git a/examples/dc3_dtlz3_3obj/trace.json b/examples/dc3_dtlz3_3obj/trace.json new file mode 100644 index 00000000..4b843ff8 --- /dev/null +++ b/examples/dc3_dtlz3_3obj/trace.json @@ -0,0 +1,42 @@ 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+]} diff --git a/examples/dc3_dtlz3_3obj/trace.py b/examples/dc3_dtlz3_3obj/trace.py new file mode 100644 index 00000000..c17c5f50 --- /dev/null +++ b/examples/dc3_dtlz3_3obj/trace.py @@ -0,0 +1,215 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: each run's population as the problem scores it, its feasible +non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +front's hypervolume with normalized objectives, in at most 40 generations; a frame's +evaluations are the runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the most infeasible solutions a frame shows +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions and the front's hypervolume + self.series = {} + + def front(self, name): + """The callback for ``run`` that records the run ``name`` after each generation, its + population scored by the problem, which the run may not optimize as it is: None without a + trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + objectives, violations = self.problem.evaluate(progress.population) + front = non_dominated(objectives[violations == 0]) + infeasible = objectives[violations > 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + history.append((progress.evaluations, front.tolist(), infeasible.tolist(), volume)) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(40) + for generation in range(max(map(len, self.series.values()), default=0)): + at = [runs[min(generation, len(runs) - 1)] for runs in self.series.values()] + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at), + "best": None, + "median": None, + "series": {name: record[3] for name, record in zip(self.series, at)}, + "state": { + "panels": [{"front": record[1], "infeasible": record[2]} for record in at] + }, + } + ) + # the front, sampled, for every panel + panel = { + "objectives": ["f1", "f2", "f3"], + "true_front": self.problem.optimal_front(600).tolist(), + } + panels = [{"title": name, "problem": panel} for name in self.series] + whole = self.normalized(self.problem.optimal_front(3_000)) + settings = { + "format": 1, + "example": "dc3_dtlz3_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": list(self.series), + }, + } + write(self.path, settings, frames.to_list()) + + +def non_dominated(points): + """The points that no other point dominates, in their order.""" + keep = [ + not np.any(np.all(points <= point, axis=1) & np.any(points < point, axis=1)) + for point in points + ] + return points[np.array(keep, dtype=bool)] if len(points) else points + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/dc3_dtlz3_3obj/trace.rs b/examples/dc3_dtlz3_3obj/trace.rs new file mode 100644 index 00000000..3fc67b81 --- /dev/null +++ b/examples/dc3_dtlz3_3obj/trace.rs @@ -0,0 +1,307 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: each run's population as the problem scores it, its feasible +//! non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +//! front's hypervolume with normalized objectives, in at most 40 generations; a frame's +//! evaluations are the runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::MultiSnapshot; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{Dc3Dtlz3, MultiProblem}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, and the +// hypervolume +type Record = (u64, Vec<[f64; 3]>, Vec<[f64; 3]>, f64); + +// the most infeasible solutions a frame shows +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation: its population scored by + // `problem`, which the run may not optimize as it is + pub fn front( + &mut self, + name: &'static str, + problem: Dc3Dtlz3<3>, + ) -> impl FnMut(&MultiSnapshot<'_, Reals, 3>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let population = snapshot.population().as_slice(); + let scores: Vec<([f64; 3], f64)> = population + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = + scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let front = non_dominated(&feasible); + let infeasible = scores.iter().filter(|s| s.1 > 0.0).map(|s| s.0); + let infeasible: Vec<[f64; 3]> = infeasible.take(SHOWN).collect(); + let volume = hypervolume(&normalized(&problem, &front), &REFERENCE, &[Minimize; 3]); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &Dc3Dtlz3<3>) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(40); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let (mut panels, mut volumes, mut evaluations) = (Vec::new(), Map::new(), 0); + for (name, history) in &self.series { + let (done, front, infeasible, volume) = &history[generation.min(history.len() - 1)]; + panels.push(json!({ "front": front, "infeasible": infeasible })); + volumes.insert(name.to_string(), json!(volume)); + evaluations += done; + } + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "series": volumes, + "state": { "panels": panels }, + })); + } + // the front, sampled, for every panel + let panel = json!({ + "objectives": ["f1", "f2", "f3"], + "true_front": problem.optimal_front(600).expect("known"), + }); + let panels: Vec = self + .series + .iter() + .map(|(name, _)| json!({ "title": name, "problem": panel })) + .collect(); + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(3_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "dc3_dtlz3_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": names, + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the points that no other point dominates, in their order +fn non_dominated(points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let dominates = + |p: &[f64; 3], q: &[f64; 3]| (0..3).all(|j| p[j] <= q[j]) && (0..3).any(|j| p[j] < q[j]); + let kept = points + .iter() + .filter(|q| !points.iter().any(|p| dominates(p, q))); + kept.copied().collect() +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/disc_brake/README.md b/examples/disc_brake/README.md index 0966e9c1..76b48eb1 100644 --- a/examples/disc_brake/README.md +++ b/examples/disc_brake/README.md @@ -6,7 +6,7 @@ reference: "Osyczka, A. and Kundu, S. (1995). A new method to solve generalized reference_url: https://doi.org/10.1007/BF01743536 optimum: "not known in closed form; from 0.1274 kg stopping in 16.654925 s to 2.0710401 s at 2.793 kg, both ends derived; genoxide's reference front has a hypervolume of 1.0853 in scaled objectives (reference point (1.1, 1.1))" languages: [rust, python] -order: 222 +order: 238 --- # Disc brake diff --git a/examples/dtlz8_3obj/README.md b/examples/dtlz8_3obj/README.md new file mode 100644 index 00000000..c7fe350b --- /dev/null +++ b/examples/dtlz8_3obj/README.md @@ -0,0 +1,62 @@ +--- +title: DTLZ8 with 3 objectives +category: multi-objective +summary: Minimize three objectives over 30 variables subject to 3 constraints, whose Pareto front is a line and a triangle, with NSGA-II and SMS-EMOA. +reference: "Deb, K., Thiele, L., Laumanns, M. and Zitzler, E. (2001). Scalable Test Problems for Evolutionary Multi-Objective Optimization. TIK-Report 112, Computer Engineering and Networks Laboratory, ETH Zürich." +reference_url: https://sop.tik.ee.ethz.ch/publicationListFiles/dtlz2001a.pdf +optimum: "the line f₁ = f₂ = t, f₃ = 1 − 4t, t in [0, 1/6], and the triangle 2f₃ + f₁ + f₂ = 1 with f₁, f₂ ≥ (1 − f₃)/4; ideal point (0, 0, 0), nadir point (3/4, 3/4, 1); hypervolume 0.9724 (normalized objectives, reference point (1.1, 1.1, 1.1))" +languages: [rust, python] +order: 199 +family: DTLZ +tab: DTLZ8 +--- + +# DTLZ8 with 3 objectives + +## The problem + +Deb, Thiele, Laumanns and Zitzler's report (2001) builds its scalable test problems three ways; the last, the constraint surface approach, gives DTLZ8 and DTLZ9: each objective is a function of its own block of variables, so that the objectives can take values independently, and constraints on the objectives cut the front out of their space. (The report's DTLZ8 is the 2002 conference paper's DTLZ7, and its DTLZ9 isn't in the paper; genoxide follows the report.) + +DTLZ8 (section 8.8, eq. 28), with 3 objectives and 30 variables in [0, 1]: + +```text +minimize fⱼ = (1/10) Σ xᵢ over the j-th block of 10 variables, j = 1, 2, 3 +subject to f₃ + 4f₁ − 1 ≥ 0 + f₃ + 4f₂ − 1 ≥ 0 + 2f₃ + f₁ + f₂ − 1 ≥ 0 +``` + +The report prints the blocks from ⌊(j − 1) n/M⌋ to ⌊j n/M⌋, which with variables counted from 1 would start at a variable x₀ and let neighbouring blocks share one: genoxide reads them as ⌊(j − 1) n/M⌋ + 1 to ⌊j n/M⌋, blocks of 10. (With more objectives, the last constraint takes the least sum of two of the first M − 1 objectives.) + +The report's front is "a combination of a straight line and a hyper-plane": the line where the first two constraints meet, f₁ = f₂ = t and f₃ = 1 − 4t for t in [0, 1/6], from (0, 0, 1) to (1/6, 1/6, 1/3), and from its end the part of the plane 2f₃ + f₁ + f₂ = 1 that the first two constraints allow, the triangle from (1/6, 1/6, 1/3) to (1/4, 3/4, 0) and (3/4, 1/4, 0). Its ideal point is (0, 0, 0) and its nadir point (3/4, 3/4, 1). The report describes the front but doesn't write it out; genoxide derives it from the constraints, and its tests check it against random feasible solutions. + +## What makes it hard + +The line and the surfaces next to it. A solution beside the line, with f₁ ≠ f₂ and f₃ just above 1 − 4 min(f₁, f₂), is only weakly dominated by the line's point: Pareto dominance keeps it, and a population fills the surfaces around the line rather than the line itself, what the report shows in its figures 28 and 29 and discusses in its figure 32. The triangle is easier: it's a plane, and every point of it is reached with its blocks at their means. + +## Representation + +A `Real` genome of 30 genes in [0, 1], the report's n = 10M. The problem is genoxide's `Dtlz8::<3>`, whose fitness is the three objectives and the total constraint violation, 0 when it is feasible; in Python, `gx.problems.Dtlz8()`. Solutions compare by constrained dominance: a feasible solution beats an infeasible one, of two infeasible ones the smaller violation wins, and of two feasible ones Pareto dominance decides. + +## Algorithm + +Two runs, with a population of 100, simulated binary crossover with η = 20 and polynomial mutation with η = 20 at a rate of 1/30 per gene: + +- NSGA-II (Deb, Pratap, Agarwal and Meyarivan, 2002, IEEE Transactions on Evolutionary Computation 6(2): 182-197), crossover at a rate of 0.9, for 500 generations, as long as the report's runs (its figures 28 and 29); +- SMS-EMOA (Beume, Naujoks and Emmerich, 2007, European Journal of Operational Research 181(3): 1653-1669), which keeps the solutions that add the most hypervolume, with as many children a generation as the population, for 20,000 generations, 2,000,000 evaluations. A solution that only a point of the front weakly dominates adds almost no hypervolume next to it, and is the first to go. + +## Output + +A line per run: the size of its final front, how many of its solutions the problem finds feasible, their IGD+ and hypervolume, and the hypervolume as a share of that of a sample of the optimal front with at least as many points as the population. Then the hypervolumes of the whole front and of the sample. + +The indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 2,000 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest feasible solution of the found front, counting only the objectives in which the solution is worse. Smaller is better; a front of as many points as the population can't cover the 2,000 exactly, and the sample shows what it can. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the volume the feasible solutions dominate up to the reference point (1.1, 1.1, 1.1). Larger is better; the whole front's is computed from 3,000 of its points, and the sample's is what a front of that many points reaches. In Python, `run` evaluates the problem in Rust, so both versions print the same. + +The page plays the runs back side by side, each population as the problem scores it: its feasible non-dominated solutions, its infeasible ones (hollow, at most 100 a frame), and the optimal front, sampled. + +[The project page](https://tachsin.gr/projects/genoxide/examples/dtlz8-3obj) plays these runs back. + +## Good results + +The target: every solution feasible, and 99% of the hypervolume of the sample of 109 points of the front, about what a front of 100 solutions spread like it has. + +NSGA-II's run with seed 1 ends with 100 solutions, 100 feasible, IGD+ 0.0355, hypervolume 0.8892, 93.6% of the sample's; SMS-EMOA's with 100 solutions, 100 feasible, IGD+ 0.0254, hypervolume 0.9462, 99.5% of the sample's. Over seeds 1 to 20, NSGA-II ends with IGD+ from 0.0332 to 0.0432 and 88.6% to 95.3% of the sample's hypervolume, no run reaching the target, and SMS-EMOA with IGD+ from 0.0249 to 0.0281 and 99.5% to 100.7% of the sample's hypervolume, every run reaching the target. The IGD+ stays above the sample's own, about 0.015: the front has a line and a triangle, and the hypervolume rewards the triangle more. diff --git a/examples/dtlz8_3obj/main.py b/examples/dtlz8_3obj/main.py new file mode 100644 index 00000000..28dc1b6e --- /dev/null +++ b/examples/dtlz8_3obj/main.py @@ -0,0 +1,92 @@ +"""DTLZ8 with 3 objectives: minimize three objectives over 30 variables subject to 3 constraints, +whose front is a line and part of a plane, with NSGA-II and SMS-EMOA. + +From genoxide's problems.Dtlz8; run evaluates it in Rust. Runs NSGA-II for the report's 500 +generations and SMS-EMOA for 20,000, with a population of 100. Prints each final front's size, how +many of its solutions are feasible, their IGD+ to 2,000 points of the optimal front and their +hypervolume, with the objectives normalized by the front's ideal and nadir points, as a share of +that of a sample of the front with at least as many points as the population; then the hypervolumes +of the whole front and of the sample. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/dtlz8_3obj/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points +REFERENCE = [1.1, 1.1, 1.1] + +problem = gx.problems.Dtlz8() +ideal, nadir = problem.ideal_point, problem.nadir_point +rate = 1 / problem.dimensions + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) +# the sample: at least as many points of the optimal front as the population has, about what a +# front of that many solutions can be +sample = normalized(problem.optimal_front(100)) + + +def run(name, algorithm, fitness, generations): + """Runs ``algorithm`` on ``fitness`` for ``generations``, and prints its final front's size, + how many of its solutions the problem finds feasible, their IGD+ to 2,000 points of the + optimal front and their hypervolume, with normalized objectives, as a share of the + sample's.""" + result = algorithm.run(fitness, generations=generations, on_generation=trace.front(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"{name}, {generations} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(2_000)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + percent = 100 * volume / gx.indicators.hypervolume(sample, REFERENCE) + print( + start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}, " + f"{percent:.1f}% of the sample's" + ) + + +# NSGA-II for the report's 500 generations +algorithm = gx.Nsga2( + problem.genome, + objectives=problem.objectives, + population_size=100, + crossover=gx.SimulatedBinaryCrossover(20), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-II", algorithm, problem, 500) +# SMS-EMOA, which keeps the solutions that add the most hypervolume, for 20,000 generations +algorithm = gx.SmsEmoa( + problem.genome, + objectives=problem.objectives, + population_size=100, + crossover=gx.SimulatedBinaryCrossover(20), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("SMS-EMOA", algorithm, problem, 20_000) +whole = normalized(problem.optimal_front(3_000)) +print( + f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}; " + f"a sample of {len(sample)} of its points: " + f"{gx.indicators.hypervolume(sample, REFERENCE):.4f}" +) +trace.write() diff --git a/examples/dtlz8_3obj/main.rs b/examples/dtlz8_3obj/main.rs new file mode 100644 index 00000000..94981e9f --- /dev/null +++ b/examples/dtlz8_3obj/main.rs @@ -0,0 +1,129 @@ +//! DTLZ8 with 3 objectives: minimize three objectives over 30 variables subject to 3 constraints, +//! whose front is a line and part of a plane, with NSGA-II and SMS-EMOA. +//! +//! From genoxide's `multi::problems::Dtlz8`. Runs NSGA-II for the report's 500 generations and +//! SMS-EMOA for 20,000, with a population of 100. Prints each final front's size, how many of its +//! solutions are feasible, their IGD+ to 2,000 points of the optimal front and their hypervolume, +//! with the objectives normalized by the front's ideal and nadir points, as a share of that of a +//! sample of the front with at least as many points as the population; then the hypervolumes of the +//! whole front and of the sample. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example dtlz8_3obj +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{Dtlz8, MultiProblem}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points +pub const REFERENCE: [f64; 3] = [1.1, 1.1, 1.1]; + +fn main() -> Result<()> { + let problem = Dtlz8::<3>::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the sample: at least as many points of the optimal front as the population has, about what + // a front of that many solutions can be + let sample = normalized(&problem, &problem.optimal_front(100).expect("known")); + let rate = 1.0 / problem.variables() as f64; + // NSGA-II for the report's 500 generations + let algorithm = Nsga2::builder(problem.representation(), [Minimize; 3]) + .population_size(100) + .crossover(SimulatedBinaryCrossover::new(20.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, "NSGA-II", algorithm, problem, 500, &sample, &mut trace, + )?; + // SMS-EMOA, which keeps the solutions that add the most hypervolume, for 20,000 generations + let algorithm = SmsEmoa::builder(problem.representation(), [Minimize; 3]) + .population_size(100) + .crossover(SimulatedBinaryCrossover::new(20.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, "SMS-EMOA", algorithm, problem, 20_000, &sample, &mut trace, + )?; + let whole = normalized(&problem, &problem.optimal_front(3_000).expect("known")); + println!( + "the whole front: hypervolume {:.4}; a sample of {} of its points: {:.4}", + hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + sample.len(), + hypervolume(&sample, &REFERENCE, &[Minimize; 3]) + ); + trace.write(&problem); + Ok(()) +} + +// runs `algorithm` on `fitness` for `generations`, and prints its final front's size, how many of +// its solutions the problem finds feasible, their IGD+ to 2,000 points of the optimal front and +// their hypervolume, with normalized objectives, as a share of the sample's +fn run( + problem: &Dtlz8<3>, + name: &'static str, + algorithm: A, + fitness: F, + generations: u64, + sample: &[[f64; 3]], + trace: &mut trace::Trace, +) -> Result<()> +where + A: MultiObjectiveAlgorithm<3, Genome = Reals>, + F: MultiFitnessFunction + Sync, +{ + let mut record = trace.front(name, *problem); + let outcome = MultiEngine::new(algorithm, fitness) + .stop_when(Stop::generations(generations)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 3], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "{name}, {generations} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(2_000).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 3]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 3]); + let percent = 100.0 * volume / hypervolume(sample, &REFERENCE, &[Minimize; 3]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the \ + sample's", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &Dtlz8<3>, points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 3]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/dtlz8_3obj/output.txt b/examples/dtlz8_3obj/output.txt new file mode 100644 index 00000000..e311230c --- /dev/null +++ b/examples/dtlz8_3obj/output.txt @@ -0,0 +1,3 @@ +NSGA-II, 500 generations: 100 solutions, 100 feasible, IGD+ 0.0355, hypervolume 0.8892, 93.6% of the sample's +SMS-EMOA, 20000 generations: 100 solutions, 100 feasible, IGD+ 0.0254, hypervolume 0.9462, 99.5% of the sample's +the whole front: hypervolume 0.9724; a sample of 109 of its points: 0.9505 diff --git a/examples/dtlz8_3obj/trace.json b/examples/dtlz8_3obj/trace.json new file mode 100644 index 00000000..e205a923 --- /dev/null +++ b/examples/dtlz8_3obj/trace.json @@ -0,0 +1,42 @@ 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+]} diff --git a/examples/dtlz8_3obj/trace.py b/examples/dtlz8_3obj/trace.py new file mode 100644 index 00000000..e37406cb --- /dev/null +++ b/examples/dtlz8_3obj/trace.py @@ -0,0 +1,215 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: each run's population as the problem scores it, its feasible +non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +front's hypervolume with normalized objectives, in at most 40 generations; a frame's +evaluations are the runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the most infeasible solutions a frame shows +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions and the front's hypervolume + self.series = {} + + def front(self, name): + """The callback for ``run`` that records the run ``name`` after each generation, its + population scored by the problem, which the run may not optimize as it is: None without a + trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + objectives, violations = self.problem.evaluate(progress.population) + front = non_dominated(objectives[violations == 0]) + infeasible = objectives[violations > 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + history.append((progress.evaluations, front.tolist(), infeasible.tolist(), volume)) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(40) + for generation in range(max(map(len, self.series.values()), default=0)): + at = [runs[min(generation, len(runs) - 1)] for runs in self.series.values()] + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at), + "best": None, + "median": None, + "series": {name: record[3] for name, record in zip(self.series, at)}, + "state": { + "panels": [{"front": record[1], "infeasible": record[2]} for record in at] + }, + } + ) + # the front, sampled, for every panel + panel = { + "objectives": ["f1", "f2", "f3"], + "true_front": self.problem.optimal_front(600).tolist(), + } + panels = [{"title": name, "problem": panel} for name in self.series] + whole = self.normalized(self.problem.optimal_front(3_000)) + settings = { + "format": 1, + "example": "dtlz8_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": list(self.series), + }, + } + write(self.path, settings, frames.to_list()) + + +def non_dominated(points): + """The points that no other point dominates, in their order.""" + keep = [ + not np.any(np.all(points <= point, axis=1) & np.any(points < point, axis=1)) + for point in points + ] + return points[np.array(keep, dtype=bool)] if len(points) else points + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/dtlz8_3obj/trace.rs b/examples/dtlz8_3obj/trace.rs new file mode 100644 index 00000000..22507820 --- /dev/null +++ b/examples/dtlz8_3obj/trace.rs @@ -0,0 +1,307 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: each run's population as the problem scores it, its feasible +//! non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +//! front's hypervolume with normalized objectives, in at most 40 generations; a frame's +//! evaluations are the runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::MultiSnapshot; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{Dtlz8, MultiProblem}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, and the +// hypervolume +type Record = (u64, Vec<[f64; 3]>, Vec<[f64; 3]>, f64); + +// the most infeasible solutions a frame shows +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation: its population scored by + // `problem`, which the run may not optimize as it is + pub fn front( + &mut self, + name: &'static str, + problem: Dtlz8<3>, + ) -> impl FnMut(&MultiSnapshot<'_, Reals, 3>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let population = snapshot.population().as_slice(); + let scores: Vec<([f64; 3], f64)> = population + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = + scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let front = non_dominated(&feasible); + let infeasible = scores.iter().filter(|s| s.1 > 0.0).map(|s| s.0); + let infeasible: Vec<[f64; 3]> = infeasible.take(SHOWN).collect(); + let volume = hypervolume(&normalized(&problem, &front), &REFERENCE, &[Minimize; 3]); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &Dtlz8<3>) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(40); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let (mut panels, mut volumes, mut evaluations) = (Vec::new(), Map::new(), 0); + for (name, history) in &self.series { + let (done, front, infeasible, volume) = &history[generation.min(history.len() - 1)]; + panels.push(json!({ "front": front, "infeasible": infeasible })); + volumes.insert(name.to_string(), json!(volume)); + evaluations += done; + } + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "series": volumes, + "state": { "panels": panels }, + })); + } + // the front, sampled, for every panel + let panel = json!({ + "objectives": ["f1", "f2", "f3"], + "true_front": problem.optimal_front(600).expect("known"), + }); + let panels: Vec = self + .series + .iter() + .map(|(name, _)| json!({ "title": name, "problem": panel })) + .collect(); + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(3_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "dtlz8_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": names, + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the points that no other point dominates, in their order +fn non_dominated(points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let dominates = + |p: &[f64; 3], q: &[f64; 3]| (0..3).all(|j| p[j] <= q[j]) && (0..3).any(|j| p[j] < q[j]); + let kept = points + .iter() + .filter(|q| !points.iter().any(|p| dominates(p, q))); + kept.copied().collect() +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/dtlz9_3obj/README.md b/examples/dtlz9_3obj/README.md new file mode 100644 index 00000000..9511c8cd --- /dev/null +++ b/examples/dtlz9_3obj/README.md @@ -0,0 +1,61 @@ +--- +title: DTLZ9 with 3 objectives +category: multi-objective +summary: Minimize three objectives over 30 variables subject to 2 constraints, whose Pareto front is a curve in a thin region, with NSGA-II and SMS-EMOA. +reference: "Deb, K., Thiele, L., Laumanns, M. and Zitzler, E. (2001). Scalable Test Problems for Evolutionary Multi-Objective Optimization. TIK-Report 112, Computer Engineering and Networks Laboratory, ETH Zürich." +reference_url: https://sop.tik.ee.ethz.ch/publicationListFiles/dtlz2001a.pdf +optimum: "the curve f₁ = f₂ = cos θ, f₃ = sin θ, θ in [0, π/2]; ideal point (0, 0, 0), nadir point (1, 1, 1); hypervolume 0.2697 (normalized objectives, reference point (1.1, 1.1, 1.1))" +languages: [rust, python] +order: 200 +family: DTLZ +tab: DTLZ9 +--- + +# DTLZ9 with 3 objectives + +## The problem + +Deb, Thiele, Laumanns and Zitzler's report (2001) builds its scalable test problems three ways; the last, the constraint surface approach, gives DTLZ8 and DTLZ9: each objective is a function of its own block of variables, so that the objectives can take values independently, and constraints on the objectives cut the front out of their space. (The report's DTLZ8 is the 2002 conference paper's DTLZ7, and its DTLZ9 isn't in the paper; genoxide follows the report.) + +DTLZ9 (section 8.9, eq. 29), with 3 objectives and 30 variables in [0, 1]: + +```text +minimize fⱼ = Σ xᵢ^0.1 over the j-th block of 10 variables, j = 1, 2, 3 +subject to f₃² + f₁² − 1 ≥ 0 + f₃² + f₂² − 1 ≥ 0 +``` + +The blocks are DTLZ8's (see its page). Unlike DTLZ8's, the objectives are sums, from 0 to 10. + +The front is the curve f₁ = f₂ = cos θ, f₃ = sin θ, θ in [0, π/2], where both constraints meet: for a given f₃ in [0, 1], each constraint asks fⱼ ≥ √(1 − f₃²) alone. With f₃ and either other objective it's a quarter of the unit circle. Its ideal point is (0, 0, 0) and its nadir point (1, 1, 1). The report describes the front but doesn't write it out; genoxide derives it from the constraints, and its tests check it against random feasible solutions. + +## What makes it hard + +The power 0.1 first: on the front each block sums to at most 1 over 10 variables, so each variable is about (fⱼ/10)¹⁰, below 10⁻¹⁰, or exactly 0, and a random solution has its objectives near the top of their range: "the density of solutions gets thinner towards the Pareto-optimal region". Then the curve: as on DTLZ8's line, a solution with f₁ ≠ f₂ next to it is only weakly dominated, and a population spreads over the surfaces around the curve (the report's figures 30 and 31). + +## Representation + +A `Real` genome of 30 genes in [0, 1], the report's n = 10M. The problem is genoxide's `Dtlz9::<3>`, whose fitness is the three objectives and the total constraint violation, 0 when it is feasible; in Python, `gx.problems.Dtlz9()`. Solutions compare by constrained dominance: a feasible solution beats an infeasible one, of two infeasible ones the smaller violation wins, and of two feasible ones Pareto dominance decides. + +## Algorithm + +Two runs, with a population of 100, simulated binary crossover with η = 20 and polynomial mutation with η = 20 at a rate of 1/30 per gene: + +- NSGA-II (Deb, Pratap, Agarwal and Meyarivan, 2002, IEEE Transactions on Evolutionary Computation 6(2): 182-197), crossover at a rate of 0.9, for 500 generations, as long as the report's runs (its figures 30 and 31); +- SMS-EMOA (Beume, Naujoks and Emmerich, 2007, European Journal of Operational Research 181(3): 1653-1669), which keeps the solutions that add the most hypervolume, with as many children a generation as the population, for 20,000 generations, 2,000,000 evaluations. A solution that only a point of the front weakly dominates adds almost no hypervolume next to it, and is the first to go. + +## Output + +A line per run: the size of its final front, how many of its solutions the problem finds feasible, their IGD+ and hypervolume, and the hypervolume as a share of that of a sample of the optimal front with at least as many points as the population. Then the hypervolumes of the whole front and of the sample. + +The indicators use the objectives normalized by the front's ideal and nadir points, so that the front spans [0, 1] in each. IGD+ (Ishibuchi et al., 2015, EMO 2015, LNCS 9019: 110-125) averages, over 2,000 points of the optimal front from genoxide's `optimal_front`, the distance to the nearest feasible solution of the found front, counting only the objectives in which the solution is worse. Smaller is better; a front of as many points as the population can't cover the 2,000 exactly, and the sample shows what it can. The hypervolume (Zitzler and Thiele, 1999, IEEE Transactions on Evolutionary Computation 3(4): 257-271) is the volume the feasible solutions dominate up to the reference point (1.1, 1.1, 1.1). Larger is better; the whole front's is computed from 3,000 of its points, and the sample's is what a front of that many points reaches. In Python, `run` evaluates the problem in Rust, so both versions print the same. + +The page plays the runs back side by side, each population as the problem scores it: its feasible non-dominated solutions, its infeasible ones (hollow, at most 100 a frame), and the optimal front, sampled. + +[The project page](https://tachsin.gr/projects/genoxide/examples/dtlz9-3obj) plays these runs back. + +## Good results + +The target: every solution feasible, and 99% of the hypervolume of the sample of 100 points of the front, about what a front of 100 solutions spread like it has. + +NSGA-II's run with seed 1 ends with 100 solutions, 100 feasible, IGD+ 3.4675, hypervolume 0.0000, 0.0% of the sample's; SMS-EMOA's with 100 solutions, 100 feasible, IGD+ 0.0024, hypervolume 0.2661, 99.9% of the sample's. Over seeds 1 to 20, NSGA-II ends with IGD+ from 3.3579 to 4.8193 and 0.0% to 0.0% of the sample's hypervolume, no run reaching the target, and SMS-EMOA with IGD+ from 0.0024 to 0.0026 and 99.8% to 99.9% of the sample's hypervolume, every run reaching the target. SMS-EMOA's IGD+ is the sample's own: its 100 solutions lie on the curve, spread along it. The report's NSGA-II and SPEA2, after 500 generations, "could not cover the entire range of the circle", with many solutions away from it. diff --git a/examples/dtlz9_3obj/main.py b/examples/dtlz9_3obj/main.py new file mode 100644 index 00000000..5cc5988d --- /dev/null +++ b/examples/dtlz9_3obj/main.py @@ -0,0 +1,92 @@ +"""DTLZ9 with 3 objectives: minimize three objectives over 30 variables subject to 2 constraints, +whose front is a curve, with NSGA-II and SMS-EMOA. + +From genoxide's problems.Dtlz9; run evaluates it in Rust. Runs NSGA-II for the report's 500 +generations and SMS-EMOA for 20,000, with a population of 100. Prints each final front's size, how +many of its solutions are feasible, their IGD+ to 2,000 points of the optimal front and their +hypervolume, with the objectives normalized by the front's ideal and nadir points, as a share of +that of a sample of the front with at least as many points as the population; then the hypervolumes +of the whole front and of the sample. + +With ``GENOXIDE_TRACE=``, it also writes a trace of its runs for the plot on the example's +page, with trace.py. + + python examples/dtlz9_3obj/main.py +""" + +import genoxide as gx + +from trace import Trace + +# the reference point of the hypervolume, with the objectives normalized by the front's ideal and +# nadir points +REFERENCE = [1.1, 1.1, 1.1] + +problem = gx.problems.Dtlz9() +ideal, nadir = problem.ideal_point, problem.nadir_point +rate = 1 / problem.dimensions + + +def normalized(points): + """The objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in + each.""" + return (points - ideal) / (nadir - ideal) + + +# with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page +trace = Trace(problem, normalized, REFERENCE) +# the sample: at least as many points of the optimal front as the population has, about what a +# front of that many solutions can be +sample = normalized(problem.optimal_front(100)) + + +def run(name, algorithm, fitness, generations): + """Runs ``algorithm`` on ``fitness`` for ``generations``, and prints its final front's size, + how many of its solutions the problem finds feasible, their IGD+ to 2,000 points of the + optimal front and their hypervolume, with normalized objectives, as a share of the + sample's.""" + result = algorithm.run(fitness, generations=generations, on_generation=trace.front(name)) + objectives, violations = problem.evaluate(result.front_genomes) + feasible = objectives[violations == 0] + noun = "solution" if len(objectives) == 1 else "solutions" + start = f"{name}, {generations} generations: {len(objectives)} {noun}, " + if len(feasible) == 0: + print(start + f"none feasible, the least violation {violations.min():.4f}") + return + optimal = normalized(problem.optimal_front(2_000)) + distance = gx.indicators.igd_plus(normalized(feasible), optimal) + volume = gx.indicators.hypervolume(normalized(feasible), REFERENCE) + percent = 100 * volume / gx.indicators.hypervolume(sample, REFERENCE) + print( + start + f"{len(feasible)} feasible, IGD+ {distance:.4f}, hypervolume {volume:.4f}, " + f"{percent:.1f}% of the sample's" + ) + + +# NSGA-II for the report's 500 generations +algorithm = gx.Nsga2( + problem.genome, + objectives=problem.objectives, + population_size=100, + crossover=gx.SimulatedBinaryCrossover(20), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("NSGA-II", algorithm, problem, 500) +# SMS-EMOA, which keeps the solutions that add the most hypervolume, for 20,000 generations +algorithm = gx.SmsEmoa( + problem.genome, + objectives=problem.objectives, + population_size=100, + crossover=gx.SimulatedBinaryCrossover(20), + mutation=gx.PolynomialMutation(20, rate=rate), + seed=1, +) +run("SMS-EMOA", algorithm, problem, 20_000) +whole = normalized(problem.optimal_front(3_000)) +print( + f"the whole front: hypervolume {gx.indicators.hypervolume(whole, REFERENCE):.4f}; " + f"a sample of {len(sample)} of its points: " + f"{gx.indicators.hypervolume(sample, REFERENCE):.4f}" +) +trace.write() diff --git a/examples/dtlz9_3obj/main.rs b/examples/dtlz9_3obj/main.rs new file mode 100644 index 00000000..fd9b9211 --- /dev/null +++ b/examples/dtlz9_3obj/main.rs @@ -0,0 +1,129 @@ +//! DTLZ9 with 3 objectives: minimize three objectives over 30 variables subject to 2 constraints, +//! whose front is a curve, with NSGA-II and SMS-EMOA. +//! +//! From genoxide's `multi::problems::Dtlz9`. Runs NSGA-II for the report's 500 generations and +//! SMS-EMOA for 20,000, with a population of 100. Prints each final front's size, how many of its +//! solutions are feasible, their IGD+ to 2,000 points of the optimal front and their hypervolume, +//! with the objectives normalized by the front's ideal and nadir points, as a share of that of a +//! sample of the front with at least as many points as the population; then the hypervolumes of the +//! whole front and of the sample. +//! +//! With `GENOXIDE_TRACE=`, it also writes a trace of its runs for the plot on the example's +//! page, with `trace.rs`. +//! +//! ```text +//! cargo run --release --example dtlz9_3obj +//! ``` + +mod trace; + +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::indicator::{hypervolume, igd_plus}; +use genoxide::multi::problems::{Dtlz9, MultiProblem}; +use genoxide::prelude::*; + +// the reference point of the hypervolume, with the objectives normalized by the front's ideal and +// nadir points +pub const REFERENCE: [f64; 3] = [1.1, 1.1, 1.1]; + +fn main() -> Result<()> { + let problem = Dtlz9::<3>::default(); + // with GENOXIDE_TRACE=, a trace of the runs for the plot on the example's page + let mut trace = trace::Trace::from_env(); + // the sample: at least as many points of the optimal front as the population has, about what + // a front of that many solutions can be + let sample = normalized(&problem, &problem.optimal_front(100).expect("known")); + let rate = 1.0 / problem.variables() as f64; + // NSGA-II for the report's 500 generations + let algorithm = Nsga2::builder(problem.representation(), [Minimize; 3]) + .population_size(100) + .crossover(SimulatedBinaryCrossover::new(20.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, "NSGA-II", algorithm, problem, 500, &sample, &mut trace, + )?; + // SMS-EMOA, which keeps the solutions that add the most hypervolume, for 20,000 generations + let algorithm = SmsEmoa::builder(problem.representation(), [Minimize; 3]) + .population_size(100) + .crossover(SimulatedBinaryCrossover::new(20.0)?) + .mutate(PolynomialMutation::per_gene(rate, 20.0)?) + .seed(1) + .build()?; + run( + &problem, "SMS-EMOA", algorithm, problem, 20_000, &sample, &mut trace, + )?; + let whole = normalized(&problem, &problem.optimal_front(3_000).expect("known")); + println!( + "the whole front: hypervolume {:.4}; a sample of {} of its points: {:.4}", + hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + sample.len(), + hypervolume(&sample, &REFERENCE, &[Minimize; 3]) + ); + trace.write(&problem); + Ok(()) +} + +// runs `algorithm` on `fitness` for `generations`, and prints its final front's size, how many of +// its solutions the problem finds feasible, their IGD+ to 2,000 points of the optimal front and +// their hypervolume, with normalized objectives, as a share of the sample's +fn run( + problem: &Dtlz9<3>, + name: &'static str, + algorithm: A, + fitness: F, + generations: u64, + sample: &[[f64; 3]], + trace: &mut trace::Trace, +) -> Result<()> +where + A: MultiObjectiveAlgorithm<3, Genome = Reals>, + F: MultiFitnessFunction + Sync, +{ + let mut record = trace.front(name, *problem); + let outcome = MultiEngine::new(algorithm, fitness) + .stop_when(Stop::generations(generations)) + .on_generation(|snapshot| record(snapshot)) + .run()?; + let scores: Vec<([f64; 3], f64)> = outcome + .front() + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let noun = if scores.len() == 1 { + "solution" + } else { + "solutions" + }; + print!( + "{name}, {generations} generations: {} {noun}, ", + scores.len() + ); + if feasible.is_empty() { + let least = scores.iter().map(|s| s.1).fold(f64::INFINITY, f64::min); + println!("none feasible, the least violation {least:.4}"); + return Ok(()); + } + let found = normalized(problem, &feasible); + let optimal = normalized(problem, &problem.optimal_front(2_000).expect("known")); + let distance = igd_plus(&found, &optimal, &[Minimize; 3]); + let volume = hypervolume(&found, &REFERENCE, &[Minimize; 3]); + let percent = 100.0 * volume / hypervolume(sample, &REFERENCE, &[Minimize; 3]); + println!( + "{} feasible, IGD+ {distance:.4}, hypervolume {volume:.4}, {percent:.1}% of the \ + sample's", + feasible.len() + ); + Ok(()) +} + +// the objectives normalized by the front's ideal and nadir points: the front spans [0, 1] in each +pub fn normalized(problem: &Dtlz9<3>, points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + let scale = |p: &[f64; 3]| std::array::from_fn(|j| (p[j] - ideal[j]) / (nadir[j] - ideal[j])); + points.iter().map(scale).collect() +} diff --git a/examples/dtlz9_3obj/output.txt b/examples/dtlz9_3obj/output.txt new file mode 100644 index 00000000..288507b7 --- /dev/null +++ b/examples/dtlz9_3obj/output.txt @@ -0,0 +1,3 @@ +NSGA-II, 500 generations: 100 solutions, 100 feasible, IGD+ 3.4675, hypervolume 0.0000, 0.0% of the sample's +SMS-EMOA, 20000 generations: 100 solutions, 100 feasible, IGD+ 0.0024, hypervolume 0.2661, 99.9% of the sample's +the whole front: hypervolume 0.2697; a sample of 100 of its points: 0.2664 diff --git a/examples/dtlz9_3obj/trace.json b/examples/dtlz9_3obj/trace.json new file mode 100644 index 00000000..b5685e07 --- /dev/null +++ b/examples/dtlz9_3obj/trace.json @@ -0,0 +1,42 @@ 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+]} diff --git a/examples/dtlz9_3obj/trace.py b/examples/dtlz9_3obj/trace.py new file mode 100644 index 00000000..0dda106c --- /dev/null +++ b/examples/dtlz9_3obj/trace.py @@ -0,0 +1,215 @@ +"""The trace of the runs for the plot on the example's page, written to the file that +``GENOXIDE_TRACE`` names: each run's population as the problem scores it, its feasible +non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +front's hypervolume with normalized objectives, in at most 40 generations; a frame's +evaluations are the runs' together. The Rust example writes the same file.""" + +import json +import math +import os + +import numpy as np + +import genoxide as gx + +# the most infeasible solutions a frame shows +SHOWN = 100 + + +class Trace: + """Records the runs through ``on_generation`` when ``GENOXIDE_TRACE`` is set.""" + + def __init__(self, problem, normalized, reference): + self.path = os.environ.get("GENOXIDE_TRACE") + self.problem = problem + self.normalized = normalized + self.reference = reference + # per run, after each generation: the evaluations, the feasible front, the infeasible + # solutions and the front's hypervolume + self.series = {} + + def front(self, name): + """The callback for ``run`` that records the run ``name`` after each generation, its + population scored by the problem, which the run may not optimize as it is: None without a + trace to record.""" + history = self.series.setdefault(name, []) + + def record(progress): + objectives, violations = self.problem.evaluate(progress.population) + front = non_dominated(objectives[violations == 0]) + infeasible = objectives[violations > 0][:SHOWN] + volume = gx.indicators.hypervolume(self.normalized(front), self.reference) + history.append((progress.evaluations, front.tolist(), infeasible.tolist(), volume)) + + return record if self.path else None + + def write(self): + """Writes the trace, if there's one: the runs side by side, a frame per generation.""" + if not self.path: + return + frames = Frames(40) + for generation in range(max(map(len, self.series.values()), default=0)): + at = [runs[min(generation, len(runs) - 1)] for runs in self.series.values()] + frames.push( + { + "generation": generation, + "evaluations": sum(record[0] for record in at), + "best": None, + "median": None, + "series": {name: record[3] for name, record in zip(self.series, at)}, + "state": { + "panels": [{"front": record[1], "infeasible": record[2]} for record in at] + }, + } + ) + # the front, sampled, for every panel + panel = { + "objectives": ["f1", "f2", "f3"], + "true_front": self.problem.optimal_front(600).tolist(), + } + panels = [{"title": name, "problem": panel} for name in self.series] + whole = self.normalized(self.problem.optimal_front(3_000)) + settings = { + "format": 1, + "example": "dtlz9_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": False, + "optimum": gx.indicators.hypervolume(whole, self.reference), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": list(self.series), + }, + } + write(self.path, settings, frames.to_list()) + + +def non_dominated(points): + """The points that no other point dominates, in their order.""" + keep = [ + not np.any(np.all(points <= point, axis=1) & np.any(points < point, axis=1)) + for point in points + ] + return points[np.array(keep, dtype=bool)] if len(points) else points + + +# ---- the same in every example's trace ---------------------------------------------------------- + + +class Frames: + """The frames of at most ``most`` generations, from the part of the run where what the page + plots changes: the frames after the last change are left out (a run that reached its target, + or a front that no longer moves), and the rest are spread evenly over the generations up to + it. While the run goes, up to 8 × ``most`` frames are kept: every ``every``-th generation, + with ``every`` doubling whenever there are that many, and the last one.""" + + def __init__(self, most): + self.most, self.every, self.kept, self.last = most, 1, [], None + + def push(self, frame): + if frame["generation"] % self.every: + self.last = frame + return + self.kept.append(frame) + self.last = None + if len(self.kept) == 8 * self.most: + self.every *= 2 + self.kept = [kept for kept in self.kept if kept["generation"] % self.every == 0] + + def to_list(self): + frames = self.kept + ([self.last] if self.last else []) + active = frames[: last_change(frames) + 1] + count, most = len(active), max(self.most, 2) + if count <= most: + return active + return [active[(i * (count - 1) + (most - 1) // 2) // (most - 1)] for i in range(most)] + + +def last_change(frames): + """The index of the frame after which nothing the page plots changes. To 3 significant + digits, as a plot shows them: the best, the median and, for a single objective (a numeric + best), the state; to within a hundredth of their range over the run: a front's + hypervolumes, in the state or in a grid's series.""" + if not frames: + return 0 + last = len(frames) - 1 + number = lambda value: isinstance(value, (int, float)) and not isinstance(value, bool) + single = any(number(frame.get("best")) for frame in frames) + + def measures(frame): + values = [] + state = frame.get("state") + hypervolume = state.get("hypervolume") if isinstance(state, dict) else None + for value in (hypervolume, frame.get("series")): + if number(value): + values.append(float(value)) + elif isinstance(value, dict): + values.extend(float(v) for _, v in sorted(value.items()) if number(v)) + return values + + measured = [measures(frame) for frame in frames] + end = measured[last] + tolerance = [] + for k in range(len(end)): + values = [values[k] for values in measured if k < len(values)] + tolerance.append((max(values) - min(values)) / 100.0) + + def same(frame, final, key, flush=False): + return coarse(frame.get(key), flush) == coarse(final.get(key), flush) + + def settled(i): + frame, final = frames[i], frames[last] + return ( + same(frame, final, "best") + and same(frame, final, "median") + and (not single or same(frame, final, "state", flush=True)) + and len(measured[i]) == len(end) + and all(abs(v - e) <= t for v, e, t in zip(measured[i], end, tolerance)) + ) + + first = last + while first > 0 and settled(first - 1): + first -= 1 + return first + + +def coarse(value, flush=False): + """``value`` with its numbers to 3 significant digits, as precisely as a plot shows them: two + frames whose plotted values agree to that precision look the same. With ``flush``, for the + solutions a plot draws on their ranges, numbers below 1e-6 in size count as 0.""" + if isinstance(value, float): + if flush and abs(value) < 1e-6: + value = 0.0 + return f"{value:.2e}" + if isinstance(value, (list, tuple)): + return "[" + ",".join(coarse(item, flush) for item in value) + "]" + if isinstance(value, dict): + items = sorted(value.items()) + return "{" + ",".join(f"{key}:{coarse(item, flush)}" for key, item in items) + "}" + return json.dumps(value) + + +def write(path, settings, frames): + """Writes the settings and the frames to ``path``, a frame per line.""" + lines = ",\n".join(map(to_json, frames)) + with open(path, "w", encoding="utf-8", newline="\n") as file: + file.write(f'{to_json(settings)[:-1]},"frames":[\n{lines}\n]}}\n') + + +def to_json(value): + """Compact JSON with sorted keys, and numbers rounded to 6 significant digits, as the Rust + example writes it.""" + return json.dumps(rounded(value), sort_keys=True, separators=(",", ":"), ensure_ascii=False) + + +def rounded(value): + if isinstance(value, dict): + return {key: rounded(item) for key, item in value.items()} + if isinstance(value, (list, tuple)): + return [rounded(item) for item in value] + if isinstance(value, float): + return float(f"{value:.5e}") if math.isfinite(value) else None + return value diff --git a/examples/dtlz9_3obj/trace.rs b/examples/dtlz9_3obj/trace.rs new file mode 100644 index 00000000..a3682541 --- /dev/null +++ b/examples/dtlz9_3obj/trace.rs @@ -0,0 +1,307 @@ +//! The trace of the runs for the plot on the example's page, written to the file that +//! `GENOXIDE_TRACE` names: each run's population as the problem scores it, its feasible +//! non-dominated solutions and its infeasible ones (at most 100), in a panel of its own, and the +//! front's hypervolume with normalized objectives, in at most 40 generations; a frame's +//! evaluations are the runs' together. The Python example writes the same file. + +use crate::{REFERENCE, normalized}; +use genoxide::Objective::Minimize; +use genoxide::multi::MultiFitnessFunction; +use genoxide::multi::MultiSnapshot; +use genoxide::multi::indicator::hypervolume; +use genoxide::multi::problems::{Dtlz9, MultiProblem}; +use genoxide::prelude::*; +use serde_json::{Map, Value, json}; + +// after a generation: the evaluations, the feasible front, the infeasible solutions, and the +// hypervolume +type Record = (u64, Vec<[f64; 3]>, Vec<[f64; 3]>, f64); + +// the most infeasible solutions a frame shows +const SHOWN: usize = 100; + +pub struct Trace { + path: Option, + // per run, its record after each generation + series: Vec<(&'static str, Vec)>, +} + +impl Trace { + // a trace for the file that GENOXIDE_TRACE names, or nothing to record if it isn't set + pub fn from_env() -> Self { + let path = std::env::var("GENOXIDE_TRACE").ok(); + Self { + path, + series: Vec::new(), + } + } + + // the callback that records the run `name` after each generation: its population scored by + // `problem`, which the run may not optimize as it is + pub fn front( + &mut self, + name: &'static str, + problem: Dtlz9<3>, + ) -> impl FnMut(&MultiSnapshot<'_, Reals, 3>) { + let tracing = self.path.is_some(); + self.series.push((name, Vec::new())); + let (_, history) = self.series.last_mut().expect("a series"); + move |snapshot| { + if !tracing { + return; + } + let population = snapshot.population().as_slice(); + let scores: Vec<([f64; 3], f64)> = population + .iter() + .map(|x| problem.evaluate(x.genome())) + .collect(); + let feasible: Vec<[f64; 3]> = + scores.iter().filter(|s| s.1 == 0.0).map(|s| s.0).collect(); + let front = non_dominated(&feasible); + let infeasible = scores.iter().filter(|s| s.1 > 0.0).map(|s| s.0); + let infeasible: Vec<[f64; 3]> = infeasible.take(SHOWN).collect(); + let volume = hypervolume(&normalized(&problem, &front), &REFERENCE, &[Minimize; 3]); + let evaluations = snapshot.progress().evaluations(); + history.push((evaluations, front, infeasible, volume)); + } + } + + // writes the trace, if there's one: the runs side by side, a frame per generation + pub fn write(self, problem: &Dtlz9<3>) { + let Some(path) = self.path else { return }; + let mut frames = Frames::new(40); + let generations = self.series.iter().map(|(_, history)| history.len()).max(); + for generation in 0..generations.unwrap_or(0) { + let (mut panels, mut volumes, mut evaluations) = (Vec::new(), Map::new(), 0); + for (name, history) in &self.series { + let (done, front, infeasible, volume) = &history[generation.min(history.len() - 1)]; + panels.push(json!({ "front": front, "infeasible": infeasible })); + volumes.insert(name.to_string(), json!(volume)); + evaluations += done; + } + frames.push(json!({ + "generation": generation, + "evaluations": evaluations, + "best": null, + "median": null, + "series": volumes, + "state": { "panels": panels }, + })); + } + // the front, sampled, for every panel + let panel = json!({ + "objectives": ["f1", "f2", "f3"], + "true_front": problem.optimal_front(600).expect("known"), + }); + let panels: Vec = self + .series + .iter() + .map(|(name, _)| json!({ "title": name, "problem": panel })) + .collect(); + let names: Vec<&str> = self.series.iter().map(|(name, _)| *name).collect(); + let whole = normalized(problem, &problem.optimal_front(3_000).expect("known")); + let settings = json!({ + "format": 1, + "example": "dtlz9_3obj", + "objective": ["minimize", "minimize", "minimize"], + "x_label": "generations", + "y_label": "hypervolume", + "log_y": false, + "optimum": hypervolume(&whole, &REFERENCE, &[Minimize; 3]), + "plot": "grid", + "problem": { + "panel_plot": "front-3d", + "panels": panels, + "series": names, + }, + }); + write(&path, settings, frames.into_vec()); + } +} + +// the points that no other point dominates, in their order +fn non_dominated(points: &[[f64; 3]]) -> Vec<[f64; 3]> { + let dominates = + |p: &[f64; 3], q: &[f64; 3]| (0..3).all(|j| p[j] <= q[j]) && (0..3).any(|j| p[j] < q[j]); + let kept = points + .iter() + .filter(|q| !points.iter().any(|p| dominates(p, q))); + kept.copied().collect() +} + +// ---- the same in every example's trace --------------------------------------------------------- + +// the frames of at most `most` generations, from the part of the run where what the page plots +// changes: the frames after the last change are left out (a run that reached its target, or a +// front that no longer moves), and the rest are spread evenly over the generations up to it. While +// the run goes, up to 8 × `most` frames are kept: every `every`-th generation, with `every` +// doubling whenever there are that many, and the last one. +struct Frames { + most: usize, + every: u64, + kept: Vec<(u64, Value)>, + last: Option<(u64, Value)>, +} + +impl Frames { + fn new(most: usize) -> Self { + let (every, kept, last) = (1, Vec::new(), None); + Self { + most, + every, + kept, + last, + } + } + + fn push(&mut self, frame: Value) { + let generation = frame["generation"].as_u64().expect("a generation"); + if !generation.is_multiple_of(self.every) { + self.last = Some((generation, frame)); + return; + } + self.kept.push((generation, frame)); + self.last = None; + if self.kept.len() == 8 * self.most { + self.every *= 2; + let every = self.every; + self.kept.retain(|(generation, _)| generation % every == 0); + } + } + + fn into_vec(self) -> Vec { + let frames = self.kept.into_iter().chain(self.last); + let frames: Vec = frames.map(|(_, frame)| frame).collect(); + let active = &frames[..=last_change(&frames)]; + let (count, most) = (active.len(), self.most.max(2)); + if count <= most { + return active.to_vec(); + } + let at = |i: usize| active[(i * (count - 1) + (most - 1) / 2) / (most - 1)].clone(); + (0..most).map(at).collect() + } +} + +// the index of the frame after which nothing the page plots changes. To 3 significant digits, as +// a plot shows them: the best, the median and, for a single objective (a numeric best), the state; +// to within a hundredth of their range over the run: a front's hypervolumes, in the state or in a +// grid's series +fn last_change(frames: &[Value]) -> usize { + let Some(last) = frames.len().checked_sub(1) else { + return 0; + }; + let single = frames.iter().any(|frame| frame["best"].is_number()); + let measures = |frame: &Value| -> Vec { + let mut values = Vec::new(); + for value in [&frame["state"]["hypervolume"], &frame["series"]] { + match value { + Value::Number(number) => values.extend(number.as_f64()), + Value::Object(map) => values.extend(map.values().filter_map(Value::as_f64)), + _ => {} + } + } + values + }; + let measured: Vec> = frames.iter().map(measures).collect(); + let end = &measured[last]; + let tolerance: Vec = (0..end.len()) + .map(|k| { + let values = measured.iter().filter_map(|values| values.get(k).copied()); + let (low, high) = values.fold((f64::INFINITY, f64::NEG_INFINITY), |(low, high), v| { + (low.min(v), high.max(v)) + }); + (high - low) / 100.0 + }) + .collect(); + let settled = |i: usize| { + let (frame, final_frame) = (&frames[i], &frames[last]); + let same = + |key: &str, flush: bool| coarse(&frame[key], flush) == coarse(&final_frame[key], flush); + same("best", false) + && same("median", false) + && (!single || same("state", true)) + && measured[i].len() == end.len() + && measured[i] + .iter() + .zip(end) + .zip(&tolerance) + .all(|((value, end), tolerance)| (value - end).abs() <= *tolerance) + }; + let mut first = last; + while first > 0 && settled(first - 1) { + first -= 1; + } + first +} + +// `value` with its numbers to 3 significant digits, as precisely as a plot shows them: two frames +// whose plotted values agree to that precision look the same. With `flush`, for the solutions a +// plot draws on their ranges, numbers below 1e-6 in size count as 0 +fn coarse(value: &Value, flush: bool) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => { + let number = number.as_f64().expect("f64"); + let number = if flush && number.abs() < 1e-6 { + 0.0 + } else { + number + }; + format!("{number:.2e}") + } + Value::Array(items) => { + format!( + "[{}]", + join(items.iter().map(|item| coarse(item, flush)).collect()) + ) + } + Value::Object(map) => { + let entry = |(key, item): (&String, &Value)| format!("{key}:{}", coarse(item, flush)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +// writes the settings and the frames to `path`, a frame per line +fn write(path: &str, settings: Value, frames: Vec) { + let frames: Vec = frames.iter().map(to_json).collect(); + let settings = to_json(&settings); + let head = &settings[..settings.len() - 1]; + let text = format!("{head},\"frames\":[\n{}\n]}}\n", frames.join(",\n")); + std::fs::write(path, text).expect("the trace is written"); +} + +// compact JSON with sorted keys, and numbers rounded to 6 significant digits and written as +// Python writes them (7542.0, 1e-08): the Python example writes the same file +fn to_json(value: &Value) -> String { + let join = |items: Vec| items.join(","); + match value { + Value::Number(number) if number.is_f64() => python_float(number.as_f64().expect("f64")), + Value::Array(items) => format!("[{}]", join(items.iter().map(to_json).collect())), + Value::Object(map) => { + let entry = + |(key, item): (&String, &Value)| format!("{}:{}", json!(key), to_json(item)); + format!("{{{}}}", join(map.iter().map(entry).collect())) + } + other => other.to_string(), + } +} + +fn python_float(value: f64) -> String { + let rounded: f64 = format!("{value:.5e}").parse().expect("a number"); + let shortest = format!("{rounded:e}"); + let (mantissa, exponent) = shortest.split_once('e').expect("an exponent"); + let exponent: i32 = exponent.parse().expect("an exponent"); + if (-4..16).contains(&exponent) { + let text = rounded.to_string(); + if text.contains('.') { + text + } else { + text + ".0" + } + } else { + let sign = if exponent < 0 { '-' } else { '+' }; + format!("{mantissa}e{sign}{:02}", exponent.abs()) + } +} diff --git a/examples/four_bar_truss/README.md b/examples/four_bar_truss/README.md index 1ba55086..f6d834d6 100644 --- a/examples/four_bar_truss/README.md +++ b/examples/four_bar_truss/README.md @@ -6,7 +6,7 @@ reference: "Stadler, W. and Dauer, J. (1993). Multicriteria optimization in engi reference_url: https://doi.org/10.2514/5.9781600866234.0209.0249 optimum: "the front from (1400, 0.04) to (2200 + 600√2, (2√2 − 2)/300) ≈ (3048.53, 0.0027614), in three pieces with x₃ = √2; hypervolume 0.8891 in objectives scaled by the ideal and nadir points (reference point (1.1, 1.1))" languages: [rust, python] -order: 224 +order: 240 --- # Four-bar truss diff --git a/examples/inverted_dtlz1/README.md b/examples/inverted_dtlz1/README.md index 93113513..0db5fb49 100644 --- a/examples/inverted_dtlz1/README.md +++ b/examples/inverted_dtlz1/README.md @@ -6,7 +6,7 @@ reference: "Jain, H. and Deb, K. (2014). An evolutionary many-objective optimiza reference_url: https://doi.org/10.1109/TEVC.2013.2281534 optimum: "the triangle f₁ + f₂ + f₃ = 1 with every fᵢ in [0, 0.5]; hypervolume 0.3392 (objectives divided by the nadir point (0.5, 0.5, 0.5), reference point (1.1, 1.1, 1.1))" languages: [rust, python] -order: 203 +order: 219 family: DTLZ tab: Inverted DTLZ1 --- diff --git a/examples/marine_design/README.md b/examples/marine_design/README.md index 869d10d5..409016f1 100644 --- a/examples/marine_design/README.md +++ b/examples/marine_design/README.md @@ -6,7 +6,7 @@ reference: "Parsons, M. G. and Scott, R. L. (2004). Formulation of multicriterio reference_url: https://doi.org/10.5957/jsr.2004.48.1.61 optimum: "not known in closed form; ideal point (8.376894, 5240.3356, −700,552.76); genoxide's reference front has a hypervolume of 0.8624 in objectives scaled by the ideal point and its estimated nadir point (11.0662, 12,435.8, −386,500) (reference point (1.1, 1.1, 1.1))" languages: [rust, python] -order: 229 +order: 245 --- # Conceptual marine design diff --git a/examples/mw1/README.md b/examples/mw1/README.md index ea114d0d..53b5bc80 100644 --- a/examples/mw1/README.md +++ b/examples/mw1/README.md @@ -6,7 +6,7 @@ reference: "Ma, Z. and Wang, Y. (2019). Evolutionary constrained multiobjective reference_url: https://doi.org/10.1109/TEVC.2019.2896967 optimum: "six pieces of the line f₂ = 1 − 0.85 f₁, from (0, 1) to (1, 0.15); hypervolume 0.6794 (normalized objectives, reference point (1.1, 1.1))" languages: [rust, python] -order: 204 +order: 220 family: MW --- diff --git a/examples/mw10/README.md b/examples/mw10/README.md index 975e3bd7..dab69aaa 100644 --- a/examples/mw10/README.md +++ b/examples/mw10/README.md @@ -6,7 +6,7 @@ reference: "Ma, Z. and Wang, Y. (2019). Evolutionary constrained multiobjective reference_url: https://doi.org/10.1109/TEVC.2019.2896967 optimum: "two pieces, each part constraint boundary and part the parabola f₂ = 1 − f₁², from (0.2325, 1.1350) to (1, 0); hypervolume 0.6883 (normalized objectives, reference point (1.1, 1.1))" languages: [rust, python] -order: 213 +order: 229 family: MW --- diff --git a/examples/mw11/README.md b/examples/mw11/README.md index ab008798..5bc85784 100644 --- a/examples/mw11/README.md +++ b/examples/mw11/README.md @@ -6,7 +6,7 @@ reference: "Ma, Z. and Wang, Y. (2019). Evolutionary constrained multiobjective reference_url: https://doi.org/10.1109/TEVC.2019.2896967 optimum: "two pieces on constraint boundaries and the isolated point (1, 1), which needs x₁ = 1 and g₃ = 1 exactly; hypervolume 0.8099 (normalized objectives, reference point (1.1, 1.1))" languages: [rust, python] -order: 214 +order: 230 family: MW --- diff --git a/examples/mw12/README.md b/examples/mw12/README.md index 53b4229b..51919463 100644 --- a/examples/mw12/README.md +++ b/examples/mw12/README.md @@ -6,7 +6,7 @@ reference: "Ma, Z. and Wang, Y. (2019). Evolutionary constrained multiobjective reference_url: https://doi.org/10.1109/TEVC.2019.2896967 optimum: "the boundary T₁ = 0 from (0, 1) to (1.3164, 0.0039); hypervolume 0.7397 (normalized objectives, reference point (1.1, 1.1))" languages: [rust, python] -order: 215 +order: 231 family: MW --- diff --git a/examples/mw13/README.md b/examples/mw13/README.md index acace1d5..f15a1836 100644 --- a/examples/mw13/README.md +++ b/examples/mw13/README.md @@ -6,7 +6,7 @@ reference: "Ma, Z. and Wang, Y. (2019). Evolutionary constrained multiobjective reference_url: https://doi.org/10.1109/TEVC.2019.2896967 optimum: "three pieces, of the unconstrained front and of the boundary T₁ = 0, from (0, 4) to (1.5, 0.0183); hypervolume 0.5819 (normalized objectives, reference point (1.1, 1.1))" languages: [rust, python] -order: 216 +order: 232 family: MW --- diff --git a/examples/mw14/README.md b/examples/mw14/README.md index 33b6272d..123cb662 100644 --- a/examples/mw14/README.md +++ b/examples/mw14/README.md @@ -6,7 +6,7 @@ reference: "Ma, Z. and Wang, Y. (2019). Evolutionary constrained multiobjective reference_url: https://doi.org/10.1109/TEVC.2019.2896967 optimum: "f₃ = (φ(f₁) + φ(f₂))/2, φ(t) = 6 − eᵗ − 1.5 sin(1.1πt²), with f₁ and f₂ each in [0, 0.7314] or (1.3296, 1.5]; hypervolume 0.6742 (objectives normalized by the ideal and nadir points, reference point (1.1, 1.1, 1.1))" languages: [rust, python] -order: 217 +order: 233 family: MW --- diff --git a/examples/mw2/README.md b/examples/mw2/README.md index 8b445443..4e29a672 100644 --- a/examples/mw2/README.md +++ b/examples/mw2/README.md @@ -6,7 +6,7 @@ reference: "Ma, Z. and Wang, Y. (2019). Evolutionary constrained multiobjective reference_url: https://doi.org/10.1109/TEVC.2019.2896967 optimum: "the line f₂ = 1 − f₁ for f₁ in [0, 1]; hypervolume 0.7100 (normalized objectives, reference point (1.1, 1.1))" languages: [rust, python] -order: 205 +order: 221 family: MW --- diff --git a/examples/mw3/README.md b/examples/mw3/README.md index cab3a026..7bdbe8ef 100644 --- a/examples/mw3/README.md +++ b/examples/mw3/README.md @@ -6,7 +6,7 @@ reference: "Ma, Z. and Wang, Y. (2019). Evolutionary constrained multiobjective reference_url: https://doi.org/10.1109/TEVC.2019.2896967 optimum: "the line f₂ = 1 − f₁ and two stretches of the second constraint's boundary, from (0, 1) to (1, 0); hypervolume 0.6650 (normalized objectives, reference point (1.1, 1.1))" languages: [rust, python] -order: 206 +order: 222 family: MW --- diff --git a/examples/mw4/README.md b/examples/mw4/README.md index 4c25512e..3e62014c 100644 --- a/examples/mw4/README.md +++ b/examples/mw4/README.md @@ -6,7 +6,7 @@ reference: "Ma, Z. and Wang, Y. (2019). Evolutionary constrained multiobjective reference_url: https://doi.org/10.1109/TEVC.2019.2896967 optimum: "the triangle f₁ + f₂ + f₃ = 1 with every fᵢ ≥ 0; hypervolume 1.1577 (reference point (1.1, 1.1, 1.1))" languages: [rust, python] -order: 207 +order: 223 family: MW --- diff --git a/examples/mw5/README.md b/examples/mw5/README.md index b7ca1bca..2dd21b3e 100644 --- a/examples/mw5/README.md +++ b/examples/mw5/README.md @@ -6,7 +6,7 @@ reference: "Ma, Z. and Wang, Y. (2019). Evolutionary constrained multiobjective reference_url: https://doi.org/10.1109/TEVC.2019.2896967 optimum: "sixteen points of the unit circle and two short curves near the axes; hypervolume 0.3930 (normalized objectives, reference point (1.1, 1.1))" languages: [rust, python] -order: 208 +order: 224 family: MW --- diff --git a/examples/mw6/README.md b/examples/mw6/README.md index 67c569e0..33ad3cd3 100644 --- a/examples/mw6/README.md +++ b/examples/mw6/README.md @@ -6,7 +6,7 @@ reference: "Ma, Z. and Wang, Y. (2019). Evolutionary constrained multiobjective reference_url: https://doi.org/10.1109/TEVC.2019.2896967 optimum: "twelve pieces of the circle f₁² + f₂² = 1.21, from f₁ = 0.0163 to (1.1, 0); hypervolume 0.4043 (normalized objectives, reference point (1.1, 1.1))" languages: [rust, python] -order: 209 +order: 225 family: MW --- diff --git a/examples/mw7/README.md b/examples/mw7/README.md index 34129bde..a0a10166 100644 --- a/examples/mw7/README.md +++ b/examples/mw7/README.md @@ -6,7 +6,7 @@ reference: "Ma, Z. and Wang, Y. (2019). Evolutionary constrained multiobjective reference_url: https://doi.org/10.1109/TEVC.2019.2896967 optimum: "three pieces, of the unit circle and of the second constraint's boundary, from (0, 1.15) to (1.15, 0); hypervolume 0.5025 (normalized objectives, reference point (1.1, 1.1))" languages: [rust, python] -order: 210 +order: 226 family: MW --- diff --git a/examples/mw8/README.md b/examples/mw8/README.md index be303f82..15ce04f7 100644 --- a/examples/mw8/README.md +++ b/examples/mw8/README.md @@ -6,7 +6,7 @@ reference: "Ma, Z. and Wang, Y. (2019). Evolutionary constrained multiobjective reference_url: https://doi.org/10.1109/TEVC.2019.2896967 optimum: "the unit sphere with non-negative coordinates where arcsin f₃ is in [0, π/24], [π/8, 5π/24], [7π/24, 3π/8] or [11π/24, π/2]; hypervolume 0.7677 (reference point (1.1, 1.1, 1.1))" languages: [rust, python] -order: 211 +order: 227 family: MW --- diff --git a/examples/mw9/README.md b/examples/mw9/README.md index 484c98fa..bd0e04fc 100644 --- a/examples/mw9/README.md +++ b/examples/mw9/README.md @@ -6,7 +6,7 @@ reference: "Ma, Z. and Wang, Y. (2019). Evolutionary constrained multiobjective reference_url: https://doi.org/10.1109/TEVC.2019.2896967 optimum: "f₂ = 1 − 0.64 f₁² for f₁ in [0, 0.5868], then f₂ = 1.15² − (f₁ + 0.15)² to (1, 0); hypervolume 0.4934 (normalized objectives, reference point (1.1, 1.1))" languages: [rust, python] -order: 212 +order: 228 family: MW --- diff --git a/examples/rocket_injector/README.md b/examples/rocket_injector/README.md index 09d46569..a699aa9b 100644 --- a/examples/rocket_injector/README.md +++ b/examples/rocket_injector/README.md @@ -6,7 +6,7 @@ reference: "Vaidyanathan, R., Tucker, P. K., Papila, N. and Shyy, W. (2003). CFD reference_url: https://doi.org/10.2514/6.2003-296 optimum: "not known in closed form; ideal point (0.0088934, −0.4315, 0.00488); genoxide's reference front has a hypervolume of 0.9019 in objectives scaled by the ideal point and its estimated nadir point (1.002, 1.0965, 1.0539) (reference point (1.1, 1.1, 1.1))" languages: [rust, python] -order: 226 +order: 242 --- # Rocket injector diff --git a/examples/scaled_dtlz1/README.md b/examples/scaled_dtlz1/README.md index 12024ec5..268d5430 100644 --- a/examples/scaled_dtlz1/README.md +++ b/examples/scaled_dtlz1/README.md @@ -6,7 +6,7 @@ reference: "Deb, K. and Jain, H. (2014). An evolutionary many-objective optimiza reference_url: https://doi.org/10.1109/TEVC.2013.2281535 optimum: "the plane f₁ + f₂/10 + f₃/100 = 0.5 with every fᵢ ≥ 0; hypervolume 1.1577 (objectives divided by the nadir point (0.5, 5, 50), reference point (1.1, 1.1, 1.1))" languages: [rust, python] -order: 201 +order: 217 family: DTLZ tab: Scaled DTLZ1 --- diff --git a/examples/scaled_dtlz2/README.md b/examples/scaled_dtlz2/README.md index 991c7c8c..a06ac93e 100644 --- a/examples/scaled_dtlz2/README.md +++ b/examples/scaled_dtlz2/README.md @@ -6,7 +6,7 @@ reference: "Deb, K. and Jain, H. (2014). An evolutionary many-objective optimiza reference_url: https://doi.org/10.1109/TEVC.2013.2281535 optimum: "the ellipsoid f₁² + (f₂/10)² + (f₃/100)² = 1 with every fᵢ ≥ 0; hypervolume 0.7971 (objectives divided by the nadir point (1, 10, 100), reference point (1.1, 1.1, 1.1))" languages: [rust, python] -order: 202 +order: 218 family: DTLZ tab: Scaled DTLZ2 --- diff --git a/examples/speed_reducer_2obj/README.md b/examples/speed_reducer_2obj/README.md index 004fed5c..e4842004 100644 --- a/examples/speed_reducer_2obj/README.md +++ b/examples/speed_reducer_2obj/README.md @@ -6,7 +6,7 @@ reference: "Kurpati, A., Azarm, S. and Wu, J. (2002). Constraint handling improv reference_url: https://doi.org/10.1007/s00158-002-0178-2 optimum: "not known in closed form; from a volume of 2771.9151 at the stress limit of 1300 to a stress of 694.70574 at a volume of 5777.9203 (best known); genoxide's reference front has a hypervolume of 1.1804 in scaled objectives (reference point (1.1, 1.1))" languages: [rust, python] -order: 223 +order: 239 --- # Speed reducer, two objectives diff --git a/examples/two_bar_truss/README.md b/examples/two_bar_truss/README.md index 1b57d2fb..49a52ff6 100644 --- a/examples/two_bar_truss/README.md +++ b/examples/two_bar_truss/README.md @@ -6,7 +6,7 @@ reference: "Deb, K., Pratap, A. and Moitra, S. (2000). Mechanical component desi reference_url: https://doi.org/10.1007/3-540-45356-3_84 optimum: "the front f₁ f₂ = 400 for stresses from 10⁵ down to 4000√5, then x₂ = 0.01 and y from 2 to 3, down to 8000√10/3 ≈ 8432.74; hypervolume 1.0663 in objectives scaled by the ideal and nadir points (reference point (1.1, 1.1))" languages: [rust, python] -order: 220 +order: 236 --- # Two-bar truss diff --git a/examples/vehicle_crashworthiness/README.md b/examples/vehicle_crashworthiness/README.md index 031e0295..fd0ae7b6 100644 --- a/examples/vehicle_crashworthiness/README.md +++ b/examples/vehicle_crashworthiness/README.md @@ -6,7 +6,7 @@ reference: "Liao, X., Li, Q., Yang, X., Zhang, W. and Li, W. (2008). Multiobject reference_url: https://doi.org/10.1007/s00158-007-0163-x optimum: "not known in closed form; ideal point (1661.7078, 6.1428, 0.0394), at bounds; genoxide's reference front has a hypervolume of 1.0525 in objectives scaled by the ideal point and its estimated nadir point (1695.161, 10.736, 0.264) (reference point (1.1, 1.1, 1.1))" languages: [rust, python] -order: 227 +order: 243 --- # Vehicle crashworthiness diff --git a/examples/water_resource_planning/README.md b/examples/water_resource_planning/README.md index ae805f05..20769723 100644 --- a/examples/water_resource_planning/README.md +++ b/examples/water_resource_planning/README.md @@ -6,7 +6,7 @@ reference: "Musselman, K. and Talavage, J. (1980). A tradeoff cut approach to mu reference_url: https://doi.org/10.1287/opre.28.6.1424 optimum: "the image of every design with x₃ = 0.01 and x₁x₂ ≥ 0.00139 / 1.0306; ideal point (63840.28, 40.46, 285346.9, 183749.97, 7.22), nadir point (73450.51, 1350, 2853469.0, 6575303.1, 25000)" languages: [rust, python] -order: 228 +order: 244 --- # Water resource planning diff --git a/examples/welded_beam_2obj/README.md b/examples/welded_beam_2obj/README.md index eb0203aa..74a9b9db 100644 --- a/examples/welded_beam_2obj/README.md +++ b/examples/welded_beam_2obj/README.md @@ -6,7 +6,7 @@ reference: "Deb, K., Pratap, A. and Moitra, S. (2000). Mechanical component desi reference_url: https://doi.org/10.1007/3-540-45356-3_84 optimum: "not known in closed form; from a cost of 2.3811341 at a deflection of 0.0157592 to a deflection of 0.00043904 at a cost of 36.421245 (best known); genoxide's reference front has a hypervolume of 1.1434 in scaled objectives (reference point (1.1, 1.1))" languages: [rust, python] -order: 221 +order: 237 --- # Welded beam, two objectives diff --git a/python/README.md b/python/README.md index dbf0df67..ee3a6954 100644 --- a/python/README.md +++ b/python/README.md @@ -260,6 +260,10 @@ The constrained problems of tunable difficulty work the same way: `Ctp1` to `Ctp constrained DTLZ problems `C1Dtlz1(objectives, variables)`, `C1Dtlz3(objectives, variables, radius)`, `C2Dtlz2`, `ConvexC2Dtlz2`, `C3Dtlz1` and `C3Dtlz4`, and Ma and Wang's `Mw1` to `Mw14` (`Mw4`, `Mw8` and `Mw14` with any number of `objectives`). +So do the constrained `Dtlz8(objectives, variables)` and `Dtlz9`, Li et al.'s DC-DTLZ problems +`Dc1Dtlz1(objectives, variables, a, b)`, `Dc1Dtlz3`, `Dc2Dtlz1`, `Dc2Dtlz3`, `Dc3Dtlz1` and +`Dc3Dtlz3`, and Fan et al.'s `DasCmop1(difficulty, variables)` to `DasCmop9`, whose +`difficulty` is a triplet (η, ζ, γ) in [0, 1]³ or the number of one of the paper's sixteen. Two submodules have constrained single-objective problems, whose fitness is `(score, violation)`: `gx.problems.cec2006` has CEC 2006's `G01()` to `G24()`, and diff --git a/python/genoxide/problems/__init__.py b/python/genoxide/problems/__init__.py index a66725d3..88b99168 100644 --- a/python/genoxide/problems/__init__.py +++ b/python/genoxide/problems/__init__.py @@ -224,6 +224,23 @@ "Mw12", "Mw13", "Mw14", + "Dtlz8", + "Dtlz9", + "Dc1Dtlz1", + "Dc1Dtlz3", + "Dc2Dtlz1", + "Dc2Dtlz3", + "Dc3Dtlz1", + "Dc3Dtlz3", + "DasCmop1", + "DasCmop2", + "DasCmop3", + "DasCmop4", + "DasCmop5", + "DasCmop6", + "DasCmop7", + "DasCmop8", + "DasCmop9", ] @dataclass(frozen=True, eq=False) @@ -2944,4 +2961,428 @@ class Mw14(_ScalableMw): _type: ClassVar[str] = "mw14" +# ---- DTLZ8 and DTLZ9 (the DTLZ report) --------------------------------------------------------- + + +@dataclass(frozen=True, init=False) +class _BlockDtlz(MultiProblem[Real]): + """DTLZ8 or DTLZ9 with ``objectives`` objectives, from ``_least`` to 6, and ``variables`` + variables, at least ``objectives``; None is the report's ``10 objectives``. The field + ``objective_count`` keeps the number, as the ``objectives`` property lists the objectives.""" + + objective_count: int + variables: int | None + _least: ClassVar[int] = 2 + + def __init__(self, objectives: int = 3, variables: int | None = None) -> None: + object.__setattr__(self, "objective_count", objectives) + object.__setattr__(self, "variables", variables) + + def _describe(self) -> dict[str, Any]: + name = type(self).__name__ + objectives = _whole(f"{name}.objectives", self.objective_count, minimum=self._least) + if objectives > 6: + raise ValueError(f"{name}.objectives is at most 6, not {objectives}") + variables = self.variables + if variables is not None: + variables = _whole(f"{name}.variables", variables, minimum=objectives, maximum=2**24) + return {"type": self._type, "objectives": objectives, "variables": variables} + + +@dataclass(frozen=True, init=False) +class Dtlz8(_BlockDtlz): + """DTLZ8: each objective the mean of its own block of variables, + ``fⱼ = (1/⌊n/M⌋) Σ xᵢ`` over ``x_{⌊(j−1)n/M⌋+1}`` to ``x_{⌊jn/M⌋}``, subject to + ``f_M + 4fⱼ − 1 ≥ 0`` for j < M and ``2f_M + min_{i≠j None: + object.__setattr__(self, "objective_count", objectives) + object.__setattr__(self, "variables", variables) + object.__setattr__(self, "a", a) + object.__setattr__(self, "b", b) + + def _describe(self) -> dict[str, Any]: + name = type(self).__name__ + objectives = _whole(f"{name}.objectives", self.objective_count, minimum=2) + if objectives > 6: + raise ValueError(f"{name}.objectives is at most 6, not {objectives}") + variables = self.variables + if variables is not None: + variables = _whole(f"{name}.variables", variables, minimum=objectives, maximum=2**24) + description: dict[str, Any] = { + "type": self._type, + "objectives": objectives, + "variables": variables, + } + for setting in ("a", "b"): + value = getattr(self, setting) + if value is not None: + if isinstance(value, bool) or not isinstance(value, (int, float)): + raise TypeError(f"{name}.{setting} is a number, not {value!r}") + value = float(value) + description[setting] = value + return description + + +@dataclass(frozen=True, init=False) +class Dc1Dtlz1(_DcDtlz): + """DC1-DTLZ1: :class:`Dtlz1` subject to ``cos(aπx₁) ≥ b``, a = 3 and b = 0.5 by default. + ``Dc1Dtlz1(objectives=3, variables=None, a=None, b=None)``: None for ``objectives + 4`` + variables. + + The front (derived from the definition): the parts of DTLZ1's, ``Σ fᵢ = 1/2``, where x₁ is + feasible, in [0, 1/9] or [5/9, 7/9]: two bands, cones from the origin. Ideal point + (0, …, 0, 1/9), nadir point (7/18, …, 7/18, 1/2). + + Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for + constrained multiobjective optimization. IEEE Transactions on Evolutionary Computation 23(2): + 303-315, supplementary document, section 1.2. a and b as the + supplement gives them. + """ + + _type: ClassVar[str] = "dc1_dtlz1" + + +@dataclass(frozen=True, init=False) +class Dc1Dtlz3(_DcDtlz): + """DC1-DTLZ3: :class:`Dtlz3` subject to ``cos(aπx₁) ≥ b``, a = 3 and b = 0.5 by default. + ``Dc1Dtlz3(objectives=3, variables=None, a=None, b=None)``: None for ``objectives + 9`` + variables. + + The front: the parts of DTLZ3's, the unit sphere, where x₁ is feasible, in [0, 1/9] or + [5/9, 7/9]. Ideal point the origin, nadir point (1, …, 1, sin(7π/18)). + + Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for + constrained multiobjective optimization. IEEE Transactions on Evolutionary Computation 23(2): + 303-315, supplementary document, section 1.2. a and b as the + supplement gives them. + """ + + _type: ClassVar[str] = "dc1_dtlz3" + + +@dataclass(frozen=True, init=False) +class Dc2Dtlz1(_DcDtlz): + """DC2-DTLZ1: :class:`Dtlz1` subject to ``cos(aπg) ≥ b`` and ``exp(−g) ≥ b``, a = 3 and + b = 0.9 by default (the authors' lab's code). ``Dc2Dtlz1(objectives=3, variables=None, a=None, + b=None)``: None for ``objectives + 4`` variables. + + The front is DTLZ1's, whole; only solutions with g below 0.048 are feasible, and the + violation has local minima on the way. Ideal point the origin, nadir point (1/2, …, 1/2). + + Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for + constrained multiobjective optimization. IEEE Transactions on Evolutionary Computation 23(2): + 303-315, supplementary document, section 1.2. b from the code + of the authors' lab (EMOC), as the supplement gives none. + """ + + _type: ClassVar[str] = "dc2_dtlz1" + + +@dataclass(frozen=True, init=False) +class Dc2Dtlz3(_DcDtlz): + """DC2-DTLZ3: :class:`Dtlz3` subject to ``cos(aπg) ≥ b`` and ``exp(−g) ≥ b``, a = 3 and + b = 0.5 by default (the authors' lab's code). ``Dc2Dtlz3(objectives=3, variables=None, a=None, + b=None)``: None for ``objectives + 9`` variables. + + The front is DTLZ3's, whole; g must be in [0, 1/9] or [5/9, ln 2]. Ideal point the origin, + nadir point (1, …, 1). + + Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for + constrained multiobjective optimization. IEEE Transactions on Evolutionary Computation 23(2): + 303-315, supplementary document, section 1.2. b from the code + of the authors' lab (EMOC), as the supplement gives none. + """ + + _type: ClassVar[str] = "dc2_dtlz3" + + +@dataclass(frozen=True, init=False) +class Dc3Dtlz1(_DcDtlz): + """DC3-DTLZ1: :class:`Dtlz1` subject to ``cos(aπxⱼ) ≥ b`` for each position variable + (j < M) and ``cos(aπg) ≥ b``, a = 3 and b = 0.5 by default: M constraints. + ``Dc3Dtlz1(objectives=3, variables=None, a=None, b=None)``: None for ``objectives + 4`` + variables. + + The front: the parts of DTLZ1's where every position variable is in [0, 1/9] or [5/9, 7/9], + 2^(M−1) patches. Ideal point (0, …, 0, 1/9), nadir point (7/18 (7/9)^(M−2), …, 7/18, 1/2). + + Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for + constrained multiobjective optimization. IEEE Transactions on Evolutionary Computation 23(2): + 303-315, supplementary document, section 1.2. The supplement + constrains x₁ to x_m; genoxide constrains the M − 1 position variables, as the authors' lab's + code does and the supplement's table 2 confirms. + """ + + _type: ClassVar[str] = "dc3_dtlz1" + + +@dataclass(frozen=True, init=False) +class Dc3Dtlz3(_DcDtlz): + """DC3-DTLZ3: :class:`Dtlz3` subject to ``cos(aπxⱼ) ≥ b`` for each position variable + (j < M) and ``cos(aπg) ≥ b``, a = 3 and b = 0.5 by default: M constraints. + ``Dc3Dtlz3(objectives=3, variables=None, a=None, b=None)``: None for ``objectives + 9`` + variables. + + The front: the parts of the unit sphere where every position variable is in [0, 1/9] or + [5/9, 7/9], 2^(M−1) patches. Ideal point (cos(7π/18)^(M−1), 0, …, 0), nadir point + (1, sin(7π/18), …, sin(7π/18)). + + Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for + constrained multiobjective optimization. IEEE Transactions on Evolutionary Computation 23(2): + 303-315, supplementary document, section 1.2. The supplement + constrains x₁ to x_m; genoxide constrains the M − 1 position variables, as the authors' lab's + code does and the supplement's table 2 confirms. + """ + + _type: ClassVar[str] = "dc3_dtlz3" + + +# ---- DAS-CMOP1-9 (Fan et al., 2020) ------------------------------------------------------------- + + +@dataclass(frozen=True) +class _DasCmop(MultiProblem[Real]): + """A DAS-CMOP problem with the difficulty triplet ``difficulty``: (η, ζ, γ), each in + [0, 1], the number of one of the paper's sixteen (table 3), or None for the triplet of the + paper's figure 6; and ``variables`` variables, 30 by default.""" + + difficulty: tuple[float, float, float] | int | None = None + variables: int = 30 + _least: ClassVar[int] = 2 + + # the paper's table 3 + STANDARD: ClassVar[tuple[tuple[float, float, float], ...]] = ( + (0.25, 0.0, 0.0), + (0.0, 0.25, 0.0), + (0.0, 0.0, 0.25), + (0.25, 0.25, 0.25), + (0.5, 0.0, 0.0), + (0.0, 0.5, 0.0), + (0.0, 0.0, 0.5), + (0.5, 0.5, 0.5), + (0.75, 0.0, 0.0), + (0.0, 0.75, 0.0), + (0.0, 0.0, 0.75), + (0.75, 0.75, 0.75), + (0.0, 1.0, 0.0), + (0.5, 1.0, 0.0), + (0.0, 1.0, 0.5), + (0.5, 1.0, 0.5), + ) + + def _describe(self) -> dict[str, Any]: + name = type(self).__name__ + variables = _whole(f"{name}.variables", self.variables, minimum=self._least, maximum=2**24) + difficulty = self.difficulty + triplet: list[float] | None = None + if isinstance(difficulty, (bool, np.bool_)): + raise TypeError(f"{name}.difficulty is a triplet or 1 to 16, not {difficulty!r}") + if isinstance(difficulty, (int, np.integer)): + number = int(difficulty) + if not 1 <= number <= 16: + raise ValueError(f"{name}.difficulty is a triplet or 1 to 16, not {number}") + triplet = list(self.STANDARD[number - 1]) + elif difficulty is not None: + levels = list(difficulty) + if len(levels) != 3 or any( + isinstance(level, bool) or not isinstance(level, (int, float)) for level in levels + ): + raise TypeError(f"{name}.difficulty is three numbers, not {difficulty!r}") + triplet = [float(level) for level in levels] + if not all(0.0 <= level <= 1.0 for level in triplet): + raise ValueError(f"{name}.difficulty's levels are in [0, 1], not {difficulty!r}") + return {"type": self._type, "variables": variables, "difficulty": triplet} + + +@dataclass(frozen=True) +class DasCmop1(_DasCmop): + """DAS-CMOP1: ``f₁ = x₁ + g``, ``f₂ = 1 − x₁² + g``, ``g = Σⱼ₌₂ⁿ (xⱼ − sin(0.5πx₁))²``, + on [0, 1]ⁿ, subject to 11 constraints whose difficulty the triplet (η, ζ, γ) sets: + ``sin(20πx₁) ≥ 2η − 1`` (the front in pieces), ``(e − g)(g − 0.5) ≥ 0`` with + ``e = 0.5 − ln ζ`` (g in a band; any g at ζ = 0, 0.5 within 10⁻⁴ at ζ = 1), and nine + rotated ellipses of size ``γ/2`` that block the way. ``DasCmop1(difficulty=None, + variables=30)``: the triplet, or the number of one of the paper's sixteen, None for + (0, 0.5, 0.5), the paper's figure 6. + + The front depends on the triplet: sampled from the definition, as the paper does. + + Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). + Difficulty adjustable and scalable constrained multiobjective test problem toolkit. + Evolutionary Computation 28(3): 339-378, table 2. + """ + + _type: ClassVar[str] = "das_cmop1" + + +@dataclass(frozen=True) +class DasCmop2(_DasCmop): + """DAS-CMOP2: :class:`DasCmop1` with the convex ``f₂ = 1 − √x₁ + g``. ``DasCmop2( + difficulty=None, variables=30)``, None for (0, 0.5, 0.5), the paper's figure 6. + + Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). + Difficulty adjustable and scalable constrained multiobjective test problem toolkit. + Evolutionary Computation 28(3): 339-378, table 2. + """ + + _type: ClassVar[str] = "das_cmop2" + + +@dataclass(frozen=True) +class DasCmop3(_DasCmop): + """DAS-CMOP3: :class:`DasCmop1` with the disconnected + ``f₂ = 1 − √x₁ + 0.5 |sin(5πx₁)| + g``. ``DasCmop3(difficulty=None, variables=30)``, None + for (0, 0.5, 0.5), the paper's figure 6. + + Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). + Difficulty adjustable and scalable constrained multiobjective test problem toolkit. + Evolutionary Computation 28(3): 339-378, table 2. + """ + + _type: ClassVar[str] = "das_cmop3" + + +@dataclass(frozen=True) +class DasCmop4(_DasCmop): + """DAS-CMOP4: :class:`DasCmop1`'s shape and constraints with the multimodal + ``g = (n − 1) + Σⱼ₌₂ⁿ ((xⱼ − 0.5)² − cos(20π(xⱼ − 0.5)))``. ``DasCmop4(difficulty=None, + variables=30)``, None for (0.5, 0.5, 0.5), the paper's figure 6. The front is DAS-CMOP1's + for the same triplet. + + Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). + Difficulty adjustable and scalable constrained multiobjective test problem toolkit. + Evolutionary Computation 28(3): 339-378, table 2. + """ + + _type: ClassVar[str] = "das_cmop4" + + +@dataclass(frozen=True) +class DasCmop5(_DasCmop): + """DAS-CMOP5: :class:`DasCmop2`'s shape with :class:`DasCmop4`'s multimodal g. + ``DasCmop5(difficulty=None, variables=30)``, None for (0.5, 0.5, 0.5), the paper's figure 6. + + Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). + Difficulty adjustable and scalable constrained multiobjective test problem toolkit. + Evolutionary Computation 28(3): 339-378, table 2. + """ + + _type: ClassVar[str] = "das_cmop5" + + +@dataclass(frozen=True) +class DasCmop6(_DasCmop): + """DAS-CMOP6: :class:`DasCmop3`'s shape with :class:`DasCmop4`'s multimodal g. + ``DasCmop6(difficulty=None, variables=30)``, None for (0.5, 0.5, 0.5), the paper's figure 6. + + Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). + Difficulty adjustable and scalable constrained multiobjective test problem toolkit. + Evolutionary Computation 28(3): 339-378, table 2. + """ + + _type: ClassVar[str] = "das_cmop6" + + +@dataclass(frozen=True) +class DasCmop7(_DasCmop): + """DAS-CMOP7: three objectives, ``f₁ = x₁x₂ + g``, ``f₂ = x₂(1 − x₁) + g``, + ``f₃ = 1 − x₂ + g``, ``g = (n − 2) + Σⱼ₌₃ⁿ ((xⱼ − 0.5)² − cos(20π(xⱼ − 0.5)))``, on + [0, 1]ⁿ, subject to 7 constraints: type I on x₁ and x₂, the type II band on g, and four + spheres of radius ``γ/2`` at the unit vectors and (1, 1, 1)/√3. ``DasCmop7(difficulty=None, + variables=30)``, None for (0.5, 0.5, 0.5), the paper's figure 6. + + Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). + Difficulty adjustable and scalable constrained multiobjective test problem toolkit. + Evolutionary Computation 28(3): 339-378, table 2. + """ + + _type: ClassVar[str] = "das_cmop7" + _least: ClassVar[int] = 3 + + +@dataclass(frozen=True) +class DasCmop8(_DasCmop): + """DAS-CMOP8: three objectives on the sphere, ``f₁ = cos(0.5πx₁) cos(0.5πx₂) + g``, + ``f₂ = cos(0.5πx₁) sin(0.5πx₂) + g``, ``f₃ = sin(0.5πx₁) + g``, with :class:`DasCmop7`'s g + and constraints. ``DasCmop8(difficulty=None, variables=30)``, None for (0.5, 0.5, 0.5). + + Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). + Difficulty adjustable and scalable constrained multiobjective test problem toolkit. + Evolutionary Computation 28(3): 339-378, table 2. + """ + + _type: ClassVar[str] = "das_cmop8" + _least: ClassVar[int] = 3 + + +@dataclass(frozen=True) +class DasCmop9(_DasCmop): + """DAS-CMOP9: :class:`DasCmop8`'s shape with ``g = Σⱼ₌₃ⁿ (xⱼ − cos(0.25jπ(x₁ + x₂)/n))²``, + linked variables, and :class:`DasCmop7`'s constraints. ``DasCmop9(difficulty=None, + variables=30)``, None for (0.5, 0.5, 0.5). + + Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). + Difficulty adjustable and scalable constrained multiobjective test problem toolkit. + Evolutionary Computation 28(3): 339-378, table 2. + """ + + _type: ClassVar[str] = "das_cmop9" + _least: ClassVar[int] = 3 + + from . import cec2006, control, engineering, multi_engineering # noqa: E402 diff --git a/python/src/problems.rs b/python/src/problems.rs index 69645c2d..b572fa3b 100644 --- a/python/src/problems.rs +++ b/python/src/problems.rs @@ -442,6 +442,86 @@ pub enum MultiConfig { objectives: usize, variables: Option, }, + Dtlz8 { + objectives: usize, + variables: Option, + }, + Dtlz9 { + objectives: usize, + variables: Option, + }, + Dc1Dtlz1 { + objectives: usize, + variables: Option, + a: Option, + b: Option, + }, + Dc1Dtlz3 { + objectives: usize, + variables: Option, + a: Option, + b: Option, + }, + Dc2Dtlz1 { + objectives: usize, + variables: Option, + a: Option, + b: Option, + }, + Dc2Dtlz3 { + objectives: usize, + variables: Option, + a: Option, + b: Option, + }, + Dc3Dtlz1 { + objectives: usize, + variables: Option, + a: Option, + b: Option, + }, + Dc3Dtlz3 { + objectives: usize, + variables: Option, + a: Option, + b: Option, + }, + DasCmop1 { + variables: usize, + difficulty: Option<[f64; 3]>, + }, + DasCmop2 { + variables: usize, + difficulty: Option<[f64; 3]>, + }, + DasCmop3 { + variables: usize, + difficulty: Option<[f64; 3]>, + }, + DasCmop4 { + variables: usize, + difficulty: Option<[f64; 3]>, + }, + DasCmop5 { + variables: usize, + difficulty: Option<[f64; 3]>, + }, + DasCmop6 { + variables: usize, + difficulty: Option<[f64; 3]>, + }, + DasCmop7 { + variables: usize, + difficulty: Option<[f64; 3]>, + }, + DasCmop8 { + variables: usize, + difficulty: Option<[f64; 3]>, + }, + DasCmop9 { + variables: usize, + difficulty: Option<[f64; 3]>, + }, } // a size of at least `minimum`, or an error that names it; the constructors panic instead @@ -585,6 +665,65 @@ fn check_constrained_dtlz( } } +// the settings of a DC-DTLZ problem, checked: 2 to 6 objectives, at least as many variables, an +// a that is finite and above 0, and a b between −1 and 1 +fn check_dc_dtlz( + name: &str, + objectives: usize, + variables: Option, + a: Option, + b: Option, +) -> Result<(), String> { + check_constrained_dtlz(objectives, variables, None, None)?; + if let Some(a) = a.filter(|a| !(a.is_finite() && *a > 0.0)) { + return Err(format!("{name} needs a finite a above 0, not {a}")); + } + if let Some(b) = b.filter(|b| !(b.is_finite() && *b > -1.0 && *b < 1.0)) { + return Err(format!("{name} needs a b between -1 and 1, not {b}")); + } + Ok(()) +} + +// a DAS-CMOP problem's difficulty triplet, checked: each level in [0, 1] +fn check_difficulty(name: &str, difficulty: Option<[f64; 3]>) -> Result<(), String> { + for level in difficulty.into_iter().flatten() { + if !(0.0..=1.0).contains(&level) { + return Err(format!( + "{name}'s difficulty levels are in [0, 1], not {level}" + )); + } + } + Ok(()) +} + +// a DC-DTLZ problem with M objectives, `variables` variables (the default if None) and the +// parameters a and b (the default's if None) +macro_rules! dc_dtlz { + ($problem:ident, $variables:expr, $a:expr, $b:expr) => {{ + let default = multi::$problem::::default(); + let variables = $variables.unwrap_or(default.variables()); + let a = $a.unwrap_or(multi::$problem::::A); + let b = $b.unwrap_or(multi::$problem::::B); + Some(multi::boxed(multi::$problem::::with_parameters( + variables, a, b, + ))) + }}; +} + +// a DAS-CMOP problem with `variables` variables and the triplet `difficulty` (the paper's figure +// 6's if None) +macro_rules! das_cmop { + ($problem:ident, $objectives:literal, $variables:expr, $difficulty:expr) => {{ + let difficulty = match $difficulty { + Some([diversity, feasibility, convergence]) => { + multi::Difficulty::new(diversity, feasibility, convergence) + } + None => multi::$problem::figure_difficulty(), + }; + try_boxed::<_, $objectives, M>(multi::$problem::new($variables, difficulty)) + }}; +} + impl MultiConfig { /// The number of objectives. pub fn objectives(&self) -> usize { @@ -618,6 +757,14 @@ impl MultiConfig { Self::Mw4 { objectives, .. } | Self::Mw8 { objectives, .. } | Self::Mw14 { objectives, .. } => objectives, + Self::Dtlz8 { objectives, .. } | Self::Dtlz9 { objectives, .. } => objectives, + Self::Dc1Dtlz1 { objectives, .. } + | Self::Dc1Dtlz3 { objectives, .. } + | Self::Dc2Dtlz1 { objectives, .. } + | Self::Dc2Dtlz3 { objectives, .. } + | Self::Dc3Dtlz1 { objectives, .. } + | Self::Dc3Dtlz3 { objectives, .. } => objectives, + Self::DasCmop7 { .. } | Self::DasCmop8 { .. } | Self::DasCmop9 { .. } => 3, Self::Viennet1 {} | Self::Viennet2 {} | Self::Viennet3 {} => 3, Self::MultiCarSideImpact {} | Self::RocketInjector {} @@ -828,6 +975,133 @@ impl MultiConfig { None => Ok(()), } } + Self::Dtlz8 { + objectives, + variables: n, + } => { + if !(3..=6).contains(&objectives) { + return Err(format!( + "DTLZ8 takes 3 to 6 objectives in Python, not {objectives}" + )); + } + match n { + Some(n) => variables(n, objectives, "DTLZ8 with this many objectives"), + None => Ok(()), + } + } + Self::Dtlz9 { + objectives, + variables: n, + } => { + if !(2..=6).contains(&objectives) { + return Err(format!( + "DTLZ9 takes 2 to 6 objectives in Python, not {objectives}" + )); + } + match n { + Some(n) => variables(n, objectives, "DTLZ9 with this many objectives"), + None => Ok(()), + } + } + Self::Dc1Dtlz1 { + objectives, + variables, + a, + b, + } => check_dc_dtlz("DC1-DTLZ1", objectives, variables, a, b), + Self::Dc1Dtlz3 { + objectives, + variables, + a, + b, + } => check_dc_dtlz("DC1-DTLZ3", objectives, variables, a, b), + Self::Dc2Dtlz1 { + objectives, + variables, + a, + b, + } => check_dc_dtlz("DC2-DTLZ1", objectives, variables, a, b), + Self::Dc2Dtlz3 { + objectives, + variables, + a, + b, + } => check_dc_dtlz("DC2-DTLZ3", objectives, variables, a, b), + Self::Dc3Dtlz1 { + objectives, + variables, + a, + b, + } => check_dc_dtlz("DC3-DTLZ1", objectives, variables, a, b), + Self::Dc3Dtlz3 { + objectives, + variables, + a, + b, + } => check_dc_dtlz("DC3-DTLZ3", objectives, variables, a, b), + Self::DasCmop1 { + variables: n, + difficulty, + } => { + variables(n, 2, "DAS-CMOP1")?; + check_difficulty("DAS-CMOP1", difficulty) + } + Self::DasCmop2 { + variables: n, + difficulty, + } => { + variables(n, 2, "DAS-CMOP2")?; + check_difficulty("DAS-CMOP2", difficulty) + } + Self::DasCmop3 { + variables: n, + difficulty, + } => { + variables(n, 2, "DAS-CMOP3")?; + check_difficulty("DAS-CMOP3", difficulty) + } + Self::DasCmop4 { + variables: n, + difficulty, + } => { + variables(n, 2, "DAS-CMOP4")?; + check_difficulty("DAS-CMOP4", difficulty) + } + Self::DasCmop5 { + variables: n, + difficulty, + } => { + variables(n, 2, "DAS-CMOP5")?; + check_difficulty("DAS-CMOP5", difficulty) + } + Self::DasCmop6 { + variables: n, + difficulty, + } => { + variables(n, 2, "DAS-CMOP6")?; + check_difficulty("DAS-CMOP6", difficulty) + } + Self::DasCmop7 { + variables: n, + difficulty, + } => { + variables(n, 3, "DAS-CMOP7")?; + check_difficulty("DAS-CMOP7", difficulty) + } + Self::DasCmop8 { + variables: n, + difficulty, + } => { + variables(n, 3, "DAS-CMOP8")?; + check_difficulty("DAS-CMOP8", difficulty) + } + Self::DasCmop9 { + variables: n, + difficulty, + } => { + variables(n, 3, "DAS-CMOP9")?; + check_difficulty("DAS-CMOP9", difficulty) + } _ => Ok(()), } } @@ -1005,6 +1279,66 @@ impl MultiConfig { Self::Mw14 { variables, .. } => Some(multi::boxed( variables.map_or_else(multi::Mw14::::default, multi::Mw14::::new), )), + Self::Dtlz8 { variables, .. } => Some(multi::boxed( + variables.map_or_else(multi::Dtlz8::::default, multi::Dtlz8::::new), + )), + Self::Dtlz9 { variables, .. } => Some(multi::boxed( + variables.map_or_else(multi::Dtlz9::::default, multi::Dtlz9::::new), + )), + Self::Dc1Dtlz1 { + variables, a, b, .. + } => dc_dtlz!(Dc1Dtlz1, variables, a, b), + Self::Dc1Dtlz3 { + variables, a, b, .. + } => dc_dtlz!(Dc1Dtlz3, variables, a, b), + Self::Dc2Dtlz1 { + variables, a, b, .. + } => dc_dtlz!(Dc2Dtlz1, variables, a, b), + Self::Dc2Dtlz3 { + variables, a, b, .. + } => dc_dtlz!(Dc2Dtlz3, variables, a, b), + Self::Dc3Dtlz1 { + variables, a, b, .. + } => dc_dtlz!(Dc3Dtlz1, variables, a, b), + Self::Dc3Dtlz3 { + variables, a, b, .. + } => dc_dtlz!(Dc3Dtlz3, variables, a, b), + Self::DasCmop1 { + variables, + difficulty, + } => das_cmop!(DasCmop1, 2, variables, difficulty), + Self::DasCmop2 { + variables, + difficulty, + } => das_cmop!(DasCmop2, 2, variables, difficulty), + Self::DasCmop3 { + variables, + difficulty, + } => das_cmop!(DasCmop3, 2, variables, difficulty), + Self::DasCmop4 { + variables, + difficulty, + } => das_cmop!(DasCmop4, 2, variables, difficulty), + Self::DasCmop5 { + variables, + difficulty, + } => das_cmop!(DasCmop5, 2, variables, difficulty), + Self::DasCmop6 { + variables, + difficulty, + } => das_cmop!(DasCmop6, 2, variables, difficulty), + Self::DasCmop7 { + variables, + difficulty, + } => das_cmop!(DasCmop7, 3, variables, difficulty), + Self::DasCmop8 { + variables, + difficulty, + } => das_cmop!(DasCmop8, 3, variables, difficulty), + Self::DasCmop9 { + variables, + difficulty, + } => das_cmop!(DasCmop9, 3, variables, difficulty), // built above Self::Zdt5 { .. } => None, }; diff --git a/python/tests/test_problems.py b/python/tests/test_problems.py index c467820b..29a501eb 100644 --- a/python/tests/test_problems.py +++ b/python/tests/test_problems.py @@ -160,6 +160,15 @@ gx.problems.Mw11, gx.problems.Mw12, gx.problems.Mw13, + gx.problems.DasCmop1, + gx.problems.DasCmop2, + gx.problems.DasCmop3, + gx.problems.DasCmop4, + gx.problems.DasCmop5, + gx.problems.DasCmop6, + gx.problems.DasCmop7, + gx.problems.DasCmop8, + gx.problems.DasCmop9, gx.problems.Dtlz1, gx.problems.Dtlz2, gx.problems.Dtlz3, @@ -189,12 +198,20 @@ gx.problems.ConvexC2Dtlz2, gx.problems.C3Dtlz1, gx.problems.C3Dtlz4, + gx.problems.Dtlz8, + gx.problems.Dtlz9, + gx.problems.Dc1Dtlz1, + gx.problems.Dc1Dtlz3, + gx.problems.Dc2Dtlz1, + gx.problems.Dc2Dtlz3, + gx.problems.Dc3Dtlz1, + gx.problems.Dc3Dtlz3, ] # on bit strings, and so not in the registry of real problems BINARY_PROBLEMS = [gx.problems.Zdt5] -# gx.problems.multi_engineering, in the registry after MW13 +# gx.problems.multi_engineering, in the registry after DAS-CMOP9 MULTI_ENGINEERING = [ gx.problems.multi_engineering.TwoBarTruss, gx.problems.multi_engineering.WeldedBeam, @@ -209,8 +226,10 @@ ] # every real multi-objective problem, in the registry's order -_AFTER_MW13 = MULTI_PROBLEMS.index(gx.problems.Mw13) + 1 -MULTI_ALL = MULTI_PROBLEMS[:_AFTER_MW13] + MULTI_ENGINEERING + MULTI_PROBLEMS[_AFTER_MW13:] +_AFTER_DAS_CMOP9 = MULTI_PROBLEMS.index(gx.problems.DasCmop9) + 1 +MULTI_ALL = ( + MULTI_PROBLEMS[:_AFTER_DAS_CMOP9] + MULTI_ENGINEERING + MULTI_PROBLEMS[_AFTER_DAS_CMOP9:] +) def test_the_classes_are_the_rust_registry(): @@ -1122,6 +1141,106 @@ def test_mw_and_dtlz_variant_sizes_are_checked_in_rust_too(): gx._genoxide.problem_info(description) +# ---- batch 11: DTLZ8, DTLZ9, DC-DTLZ and DAS-CMOP ----------------------------------------------- + + +def test_dtlz8_and_dtlz9_at_chosen_points(): + # DTLZ8 (M = 3, n = 30): blocks at 1/8, 1/4 and 5/8, all feasible + x = np.repeat([0.125, 0.25, 0.625], 10) + objectives, violation = gx.problems.Dtlz8()(x) + assert list(objectives) == pytest.approx([0.125, 0.25, 0.625]) and violation == 0.0 + assert list(gx.problems.Dtlz8().constraints(x)) == pytest.approx([-0.125, -0.625, -0.625]) + front = gx.problems.Dtlz8().optimal_front(100) + assert len(front) >= 100 + assert gx.problems.Dtlz8().nadir_point == pytest.approx([0.75, 0.75, 1.0]) + # DTLZ9: a block of 10 genes at 2⁻¹⁰ sums to 5; the front on the unit circle + x = np.ones(30) + x[:10] = 0.5**10 + objectives, violation = gx.problems.Dtlz9()(x) + assert list(objectives) == pytest.approx([5.0, 10.0, 10.0]) and violation == 0.0 + front = gx.problems.Dtlz9().optimal_front(50) + assert np.allclose(front[:, 0] ** 2 + front[:, 2] ** 2, 1) and np.all(front[:, 0] == front[:, 1]) + assert gx.problems.Dtlz9(objectives=2).dimensions == 20 + assert gx.problems.Dtlz8(objectives=4).constraint_count == 4 + + +def test_dc_dtlz_at_chosen_points(): + # DC1-DTLZ1: x₁ = 1/3 breaks cos(3πx₁) ≥ 0.5 by 1.5 + x = np.full(7, 0.5) + x[0] = 1 / 3 + objectives, violation = gx.problems.Dc1Dtlz1()(x) + assert list(objectives) == pytest.approx([1 / 12, 1 / 12, 1 / 3]) + assert violation == pytest.approx(1.5) + # DC2-DTLZ1 (b = 0.9) at g = 0: feasible by 0.1 twice; with a = 2, b = 0.5 by 0.5 + assert list(gx.problems.Dc2Dtlz1().constraints(np.full(7, 0.5))) == pytest.approx([-0.1, -0.1]) + problem = gx.problems.Dc2Dtlz1(a=2, b=0.5) + assert list(problem.constraints(np.full(7, 0.5))) == pytest.approx([-0.5, -0.5]) + # DC3: M constraints + assert gx.problems.Dc3Dtlz3(objectives=5).constraint_count == 5 + assert gx.problems.Dc3Dtlz3().dimensions == 12 + # the fronts: on DTLZ's, with the derived ideal and nadir points + front = gx.problems.Dc1Dtlz1().optimal_front(91) + assert np.allclose(front.sum(axis=1), 0.5) + assert gx.problems.Dc1Dtlz1().ideal_point == pytest.approx([0, 0, 1 / 9]) + assert gx.problems.Dc1Dtlz1().nadir_point == pytest.approx([7 / 18, 7 / 18, 0.5]) + front = gx.problems.Dc3Dtlz3().optimal_front(91) + assert np.allclose((front**2).sum(axis=1), 1) + assert gx.problems.Dc2Dtlz3(objectives=2).optimal_front(10).shape == (10, 2) + + +def test_das_cmop_at_chosen_points(): + # DAS-CMOP4 at x₁ = 0.25 with the distance variables at 0.5: g = 0, on the concave front + x = np.full(30, 0.5) + x[0] = 0.25 + objectives, violation = gx.problems.DasCmop4()(x) + assert list(objectives) == pytest.approx([0.25, 0.9375]) + # the default triplet (0.5, 0.5, 0.5) asks g ≥ 0.5: broken by (e − 0)(0.5 − 0) and more + assert violation > 0 + assert gx.problems.DasCmop4().constraint_count == 11 + assert gx.problems.DasCmop7().constraint_count == 7 + # the triplet by number or by value, the same problem + first = gx.problems.DasCmop1(difficulty=8).optimal_front(50) + assert np.array_equal(first, gx.problems.DasCmop1(difficulty=(0.5, 0.5, 0.5)).optimal_front(50)) + # with no constraint, the unconstrained front f₂ = 1 − f₁² + front = gx.problems.DasCmop1(difficulty=(0, 0, 0)).optimal_front(101) + assert np.allclose(front[:, 1], 1 - front[:, 0] ** 2) + front = gx.problems.DasCmop8(difficulty=(0, 0, 0)).optimal_front(91) + assert len(front) >= 91 and np.allclose((front**2).sum(axis=1), 1) + assert gx.problems.DasCmop9(variables=10).dimensions == 10 + + +@pytest.mark.parametrize( + "problem, message", + [ + (gx.problems.Dtlz8(objectives=2), "Dtlz8.objectives is at least 3, not 2"), + (gx.problems.Dtlz9(objectives=7), "Dtlz9.objectives is at most 6, not 7"), + (gx.problems.Dtlz9(variables=2), "Dtlz9.variables is at least 3, not 2"), + (gx.problems.Dc1Dtlz1(a=0), "DC1-DTLZ1 needs a finite a above 0, not 0"), + (gx.problems.Dc2Dtlz3(b=1), "DC2-DTLZ3 needs a b between -1 and 1, not 1"), + (gx.problems.Dc3Dtlz1(a="a"), "Dc3Dtlz1.a is a number"), + (gx.problems.DasCmop1(difficulty=17), "DasCmop1.difficulty is a triplet or 1 to 16"), + (gx.problems.DasCmop2(difficulty=(0, 2, 0)), "DasCmop2.difficulty's levels are in"), + (gx.problems.DasCmop3(difficulty=(0, 0)), "DasCmop3.difficulty is three numbers"), + (gx.problems.DasCmop7(variables=2), "DasCmop7.variables is at least 3, not 2"), + ], +) +def test_wrong_batch_11_settings_are_errors(problem, message): + with pytest.raises((ValueError, TypeError), match=message): + problem.genome + + +def test_batch_11_settings_are_checked_in_rust_too(): + description = '{"type": "das_cmop1", "variables": 30, "difficulty": [0.0, 1.5, 0.0]}' + with pytest.raises(ValueError, match="DAS-CMOP1's difficulty levels are in"): + gx._genoxide.problem_info(description) + description = '{"type": "dtlz8", "objectives": 2, "variables": null}' + with pytest.raises(ValueError, match="DTLZ8 takes 3 to 6 objectives"): + gx._genoxide.problem_info(description) + description = '{"type": "dc1_dtlz1", "objectives": 3, "variables": null, "a": -1, "b": null}' + with pytest.raises(ValueError, match="DC1-DTLZ1 needs a finite a above 0"): + gx._genoxide.problem_info(description) + + def _non_dominated_2d(values): """The non-dominated rows of an (n, 2) array.""" values = values[np.lexsort((values[:, 1], values[:, 0]))] diff --git a/site/lib/projects/genoxide/examples.js b/site/lib/projects/genoxide/examples.js index 25d41ad2..6bf3668a 100644 --- a/site/lib/projects/genoxide/examples.js +++ b/site/lib/projects/genoxide/examples.js @@ -251,6 +251,8 @@ const FAMILY_SUMMARIES = { "The constrained problems of the CEC 2006 competition: linear and nonlinear inequalities and equalities, some with a feasible region that is tiny or in pieces.", CTP: "Two objectives, two variables and one or two constraints from a tunable generator: fronts cut into pieces or points, behind infeasible bands or at the ends of narrow tunnels.", "C-DTLZ": "DTLZ problems with constraints, here with three objectives: an infeasible barrier before the front, holes in it, or a new front on the constraints' boundaries.", + "DAS-CMOP": "Two or three objectives with constraints whose difficulty a triplet sets: fronts in pieces, a narrow band of feasible distances, and infeasible regions in the way.", + "DC-DTLZ": "DTLZ problems with constraints on the variables, here with three objectives: the front in cones or patches, and local optima of the constraint violation on the way.", DTLZ: "Scalable to any number of objectives, here three: fronts that are a plane, part of a sphere, a curve or disconnected regions, some behind many local fronts, and convex, scaled and inverted versions.", MW: "Two or three objectives over 15 variables with one to four constraints: fronts whole, in pieces, in points or on the constraints' boundaries, some reached only through narrow feasible regions.", WFG: "Two objectives from the WFG toolkit: convex, concave, linear, mixed and disconnected fronts behind biased, deceptive, multimodal and non-separable parameters.", diff --git a/src/multi/problems.rs b/src/multi/problems.rs index 8dbb02b5..e9534c91 100644 --- a/src/multi/problems.rs +++ b/src/multi/problems.rs @@ -40,6 +40,7 @@ //! | [`Dtlz1`], [`Dtlz2`], [`Dtlz3`], [`Dtlz4`] | M or more (M + 4, M + 9) | any M ≥ 2 | | linear; spherical | //! | [`Dtlz5`], [`Dtlz6`] | M or more (M + 9) | any M ≥ 2 | | a curve for M ≤ 3; not known for more | //! | [`Dtlz7`] | M or more (M + 19) | any M ≥ 2 | | 2^(M−1) disconnected regions | +//! | [`Dtlz8`], [`Dtlz9`] | M or more (10 M) | M ≥ 3; any M ≥ 2 | M; M − 1 | a line and part of a plane; a curve | //! | [`ConvexDtlz2`], [`ScaledDtlz2`] | M or more (M + 9) | any M ≥ 2 | | convex; a scaled sphere | //! | [`ScaledDtlz1`], [`InvertedDtlz1`] | M or more (M + 4) | any M ≥ 2 | | a scaled plane; an inverted simplex | //! | [`Wfg1`], [`Wfg2`], [`Wfg3`] | k + l (k = 4 or 2(M − 1), l = 20) | any M ≥ 2 | | convex and mixed; convex, disconnected; linear for M = 2 | @@ -61,8 +62,13 @@ //! | [`C1Dtlz1`], [`C1Dtlz3`] | M or more (M + 4, M + 9) | any M ≥ 2 | 1 | DTLZ1's, DTLZ3's, behind infeasible barriers | //! | [`C2Dtlz2`], [`ConvexC2Dtlz2`] | M or more (M + 9) | any M ≥ 2 | 1 | parts of DTLZ2's, of convex DTLZ2's | //! | [`C3Dtlz1`], [`C3Dtlz4`] | M or more (M + 4) | any M ≥ 2 | M | on the constraints' boundaries | +//! | [`Dc1Dtlz1`], [`Dc1Dtlz3`] | M or more (M + 4, M + 9) | any M ≥ 2 | 1 | DTLZ1's, DTLZ3's, in cones | +//! | [`Dc2Dtlz1`], [`Dc2Dtlz3`] | M or more (M + 4, M + 9) | any M ≥ 2 | 2 | DTLZ1's, DTLZ3's, nearly all else infeasible | +//! | [`Dc3Dtlz1`], [`Dc3Dtlz3`] | M or more (M + 4, M + 9) | any M ≥ 2 | M | DTLZ1's, DTLZ3's, in patches | //! | [`Mw1`] to [`Mw3`], [`Mw5`] to [`Mw7`], [`Mw9`] to [`Mw13`] | 3 or more (15) | 2 | 1 to 4 | disconnected, points, or on constraint boundaries | //! | [`Mw4`], [`Mw8`], [`Mw14`] | M + 1 or more (M + 12) | any M ≥ 2 | 1 | linear; spherical in four bands; 2^(M−1) patches | +//! | [`DasCmop1`] to [`DasCmop6`] | 2 or more (30) | 2 | 11 | sampled; per [`Difficulty`] triplet | +//! | [`DasCmop7`] to [`DasCmop9`] | 3 or more (30) | 3 | 7 | sampled; per [`Difficulty`] triplet | //! | [`engineering`]: two-bar and four-bar trusses, welded beam, disc brake, speed reducer | 3 to 7 | 2 | 0 to 11 | the trusses' derived; the others not known | //! | [`engineering`]: car side impact, rocket injector, vehicle crashworthiness, marine design; water resource planning | 3 to 7 | 3; 5 | 0, 9 or 10; 7 | not known; derived | //! @@ -75,7 +81,10 @@ //! DTLZ problems are Deb and Jain's (2014, *IEEE Transactions on Evolutionary Computation* 18(4): //! 577-601 and 602-622), CTP Deb, Pratap and Meyarivan's (2001, EMO 2001, LNCS 1993: 284-298), //! C-DTLZ Jain and Deb's (2014, *IEEE Transactions on Evolutionary Computation* 18(4): 602-622), -//! and MW Ma and Wang's (2019, *IEEE Transactions on Evolutionary Computation* 23(6): 972-986). +//! MW Ma and Wang's (2019, *IEEE Transactions on Evolutionary Computation* 23(6): 972-986), +//! DC-DTLZ Li, Chen, Fu and Yao's (2019, *IEEE Transactions on Evolutionary Computation* 23(2): +//! 303-315), and DAS-CMOP Fan et al.'s (2020, *Evolutionary Computation* 28(3): 339-378); DTLZ8 +//! and DTLZ9 are the DTLZ report's. //! [`Zdt5`] has //! [`Binary`](crate::genome::Binary) genomes, and so isn't in [`all`], whose problems have //! [`Real`] ones. Each other problem's docs give its definition and cite its original authors. @@ -90,7 +99,10 @@ mod cdtlz; mod classic; mod ctp; +mod das_cmop; +mod dcdtlz; mod dtlz; +mod dtlz_constrained; mod dtlz_variants; pub mod engineering; mod mw; @@ -103,7 +115,13 @@ pub use classic::{ Viennet2, Viennet3, }; pub use ctp::{Ctp1, Ctp2, Ctp3, Ctp4, Ctp5, Ctp6, Ctp7, Ctp8}; +pub use das_cmop::{ + DasCmop1, DasCmop2, DasCmop3, DasCmop4, DasCmop5, DasCmop6, DasCmop7, DasCmop8, DasCmop9, + Difficulty, +}; +pub use dcdtlz::{Dc1Dtlz1, Dc1Dtlz3, Dc2Dtlz1, Dc2Dtlz3, Dc3Dtlz1, Dc3Dtlz3}; pub use dtlz::{Dtlz1, Dtlz2, Dtlz3, Dtlz4, Dtlz5, Dtlz6, Dtlz7}; +pub use dtlz_constrained::{Dtlz8, Dtlz9}; pub use dtlz_variants::{ConvexDtlz2, InvertedDtlz1, ScaledDtlz1, ScaledDtlz2}; pub use mw::{Mw1, Mw2, Mw3, Mw4, Mw5, Mw6, Mw7, Mw8, Mw9, Mw10, Mw11, Mw12, Mw13, Mw14}; pub use wfg::{Wfg1, Wfg2, Wfg3, Wfg4, Wfg5, Wfg6, Wfg7, Wfg8, Wfg9}; @@ -304,12 +322,12 @@ where } /// Every problem of this module with `M` objectives and [`Real`] genomes, at its default size: -/// the two-objective problems for `M = 2`, the Viennet problems for `M = 3`, and the +/// the two-objective problems for `M = 2`, the Viennet problems and DAS-CMOP7-9 for `M = 3`, and the /// [`engineering`] problems with `M` objectives, in the order of the table above, then DTLZ1-7, /// the convex, scaled and inverted DTLZ problems, WFG1-9, MW4, MW8 /// and MW14 for any `M` from 2 on, then the constrained DTLZ problems (C1-DTLZ3 and convex /// C2-DTLZ2 only for the numbers of objectives their paper gives a radius for: 3, 5, 8, 10 and -/// 15). [`Zdt5`], on bit strings, isn't in it. +/// 15), then DTLZ8 (from 3 objectives on), DTLZ9 and the DC-DTLZ problems. [`Zdt5`], on bit strings, isn't in it. pub fn all() -> Vec>> { let fixed = [ try_boxed::<_, 2, M>(Zdt1::default()), @@ -349,6 +367,15 @@ pub fn all() -> Vec>> { try_boxed::<_, 2, M>(Mw11::default()), try_boxed::<_, 2, M>(Mw12::default()), try_boxed::<_, 2, M>(Mw13::default()), + try_boxed::<_, 2, M>(DasCmop1::default()), + try_boxed::<_, 2, M>(DasCmop2::default()), + try_boxed::<_, 2, M>(DasCmop3::default()), + try_boxed::<_, 2, M>(DasCmop4::default()), + try_boxed::<_, 2, M>(DasCmop5::default()), + try_boxed::<_, 2, M>(DasCmop6::default()), + try_boxed::<_, 3, M>(DasCmop7::default()), + try_boxed::<_, 3, M>(DasCmop8::default()), + try_boxed::<_, 3, M>(DasCmop9::default()), try_boxed::<_, 2, M>(engineering::TwoBarTruss), try_boxed::<_, 2, M>(engineering::WeldedBeam), try_boxed::<_, 2, M>(engineering::DiscBrake), @@ -389,6 +416,13 @@ pub fn all() -> Vec>> { if M >= 2 { problems.extend(cdtlz::all::()); } + if M >= 3 { + problems.push(boxed(Dtlz8::::default())); + } + if M >= 2 { + problems.push(boxed(Dtlz9::::default())); + problems.extend(dcdtlz::all::()); + } problems } @@ -688,7 +722,7 @@ mod tests { #[test] fn the_registries_describe_every_problem() { let two = all::<2>(); - assert_eq!(two.len(), 66); + assert_eq!(two.len(), 79); check_registry(two); let three = all::<3>(); assert_eq!( @@ -697,6 +731,9 @@ mod tests { "VNT1", "VNT2", "VNT3", + "DAS-CMOP7", + "DAS-CMOP8", + "DAS-CMOP9", "CarSideImpact", "RocketInjector", "VehicleCrashworthiness", @@ -730,6 +767,14 @@ mod tests { "convex C2-DTLZ2", "C3-DTLZ1", "C3-DTLZ4", + "DTLZ8", + "DTLZ9", + "DC1-DTLZ1", + "DC1-DTLZ3", + "DC2-DTLZ1", + "DC2-DTLZ3", + "DC3-DTLZ1", + "DC3-DTLZ3", ] ); check_registry(three); diff --git a/src/multi/problems/das_cmop.rs b/src/multi/problems/das_cmop.rs new file mode 100644 index 00000000..501cd471 --- /dev/null +++ b/src/multi/problems/das_cmop.rs @@ -0,0 +1,1467 @@ +//! DAS-CMOP1-DAS-CMOP9: Fan et al.'s difficulty-adjustable constrained problems, two objectives +//! for DAS-CMOP1-6 and three for DAS-CMOP7-9, each with a [`Difficulty`] triplet. +//! +//! Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and Goodman, E. (2020). +//! Difficulty adjustable and scalable constrained multiobjective test problem toolkit. +//! *Evolutionary Computation* 28(3): 339-378, sections 4-6 (eqs. 3-7, tables 2 and 3, figure 6), +//! read in arXiv:1612.07603v3 (the journal's text wasn't compared; the arXiv's first version had +//! other problems). +//! +//! Every problem is `fᵢ = αᵢ(x_I) + g(x_II)`: shape functions α of the position variables (x₁ for +//! two objectives, x₁ and x₂ for three) give the unconstrained front, at g = 0, and a distance +//! function g ≥ 0 of the others moves a solution away from it along the diagonal (1, …, 1). Three +//! kinds of constraints each set one kind of difficulty, and the triplet (η, ζ, γ) in [0, 1]³ its +//! level: +//! +//! - **type I, diversity (η):** `sin(aπx₁) − b ≥ 0` (and `cos(aπx₂) − b ≥ 0` for three +//! objectives), with a = 20 and `b = 2η − 1`: the front in 10 or more pieces, narrower as η +//! grows; none at η = 0; +//! - **type II, feasibility (ζ):** `(e − g)(g − d) ≥ 0`, with d = 0.5 and `e = d − ln ζ`: g, the +//! distance from the unconstrained front, must be between d and e, a band of feasible solutions +//! that narrows as ζ grows; at ζ = 0 (where the paper sets d = 0 and e is infinite) any g, and +//! at ζ = 1 (e = d) exactly g = 0.5, an equality; +//! - **type III, convergence (γ):** infeasible regions in the objective space, of size `r = γ/2`, +//! that block the way to the front: nine ellipses for two objectives, `((f₁ − p_k) cos θ − +//! (f₂ − q_k) sin θ)²/a_k² + ((f₁ − p_k) sin θ + (f₂ − q_k) cos θ)²/b_k² ≥ r` with +//! `a_k² = 0.3`, `b_k² = 1.2`, θ = −π/4 and centers (p_k, q_k) along the anti-diagonals, +//! and four spheres for three objectives, `Σⱼ (fⱼ − c_kⱼ)² ≥ r²` with centers at the unit +//! vectors and at (1, 1, 1)/√3. +//! +//! [`Difficulty::STANDARD`] has the paper's sixteen triplets (table 3). +//! +//! **What was checked where.** The formulas, constants and triplets are the paper's table 2 and +//! 3, compared with the authors' Java code (the jMetal classes `DASCMOP1` to `DASCMOP9` in their +//! "Java source codes of MOEA/D-CDP, NSGA-II-CDP, C-MOEA/DD and C-NSGA-III", linked from the +//! paper and hosted by the first author's laboratory at +//! ; read only, to compare: +//! it has no license). They agree but for these points, where genoxide follows the code: +//! +//! - table 2 sums DAS-CMOP1-3's g from j = 1, which would add `(x₁ − sin(0.5πx₁))²`, a term of +//! the position variable; the code sums from j = 2, as the paper's own examples (eqs. 3-5) do; +//! - table 2's `a_k² = 0.3, b_k² = 1.2` are squares, which the code divides by (eq. 5's example +//! has 0.4 and 1.6): the ellipses' semi-axes are √(0.3 r) and √(1.2 r); +//! - at ζ = 1 the code accepts `|g − 0.5| ≤ 10⁻⁴`, where the paper's product is feasible only at +//! g = 0.5 exactly; genoxide does the same, and [`constraints`](MultiProblem::constraints) +//! gives `|g − 0.5| − 10⁻⁴` there. At ζ = 0 it gives `−g`, always feasible, the limit of the +//! product divided by e as e grows (the code sets e = 10³⁰). +//! +//! [`constraints`](MultiProblem::constraints) gives the constraints in the paper's order as +//! `g(x) <= 0`: `b − sin(aπx₁)` (then `b − cos(aπx₂)`), the type II constraint `−(e − g)(g − d)`, +//! then `r − (the ellipse's form)` for each of the nine ellipses or `r² − Σⱼ (fⱼ − c_kⱼ)²` for +//! each of the four spheres. +//! +//! **The fronts.** The paper samples the unconstrained front and the constraints' boundaries and +//! filters them (its figure 6, and the files the authors published, 1,000 points for two +//! objectives and 10,000 for three); genoxide derives them the same way, from the definition. +//! Each value of the position variables gives a ray in the objective space, `α + g (1, …, 1)`, +//! and going further along a ray only makes a solution worse: the front is made of the first +//! feasible point of each ray where the type I constraints hold, the least g in the type II band +//! outside the ellipses or spheres (found exactly: along a ray each is a quadratic in g), then +//! non-dominated. That's 20,000 rays evenly spread over the feasible x₁, with the ends of its +//! intervals, for two objectives, and the 41,905 rays of Das and Dennis's points with 288 +//! divisions, mapped to x₁ and x₂, for three; for three objectives `optimal_front` takes the rays +//! of fewer divisions, a divisor of 288, and the points of the dense front where each objective +//! is least and largest, the ideal and nadir points. The fronts depend on the triplet, not on the +//! number of variables, as long as g reaches the values they need (under 1 for the paper's +//! triplets; the default 30 variables reach far more). +//! +//! Compared with the fronts the authors published (`pf_data.zip`, the 16 triplets of each problem; +//! compared only), the fronts here agree to within their sampling: the mean distance from each +//! of their points to the nearest point here is under 0.01 for 130 of the 144 files, and from +//! each point here to the nearest of theirs under 0.01 for 140. The rest are points of theirs +//! that break the constraints as the paper and the code write them, and points here that they +//! leave out: on DAS-CMOP2 and DAS-CMOP5 with the triplets 12, 15 and 16, their points with g +//! other than 0.5 under ζ = 1, or with x₁ outside the type I intervals; on DAS-CMOP3 and +//! DAS-CMOP6 with η = 0.5 (triplets 5, 8, 14 and 16), the isolated point x₁ = 1, feasible only in +//! exact arithmetic (`sin 20π` is −2.4 × 10⁻¹⁵ in floating point); and on DAS-CMOP8 and +//! DAS-CMOP9 with the triplets 7 and 11 (γ alone), the parts of the front near the corners that +//! the type III spheres push out along the diagonal, which their files leave out (the unit +//! sphere's points there are infeasible, and these, further out, aren't dominated by any +//! feasible point). + +use super::mw::spread_by_length; +use super::{MultiProblem, das_dennis, divisions_for, non_dominated}; +use crate::constraint::at_most; +use crate::genome::{Real, Reals}; +use crate::math; +use crate::multi::MultiFitnessFunction; +use crate::problems::Constraints; +use std::cmp::Ordering; +use std::collections::{BTreeMap, HashMap}; +use std::f64::consts::PI; +use std::sync::{Arc, Mutex, OnceLock}; + +const REFERENCE: &str = "Fan, Z., Li, W., Cai, X., Li, H., Wei, C., Zhang, Q., Deb, K. and \ + Goodman, E. (2020). Difficulty adjustable and scalable constrained \ + multiobjective test problem toolkit. Evolutionary Computation 28(3): \ + 339-378."; +const REFERENCE_URL: &str = "https://doi.org/10.1162/evco_a_00259"; + +// the paper's number of variables +const VARIABLES: usize = 30; + +// the type I constraints' a, and the type II constraints' d +const A: f64 = 20.0; +const D: f64 = 0.5; + +// the tolerance of the type II equality at ζ = 1, the authors' code's +const TOLERANCE: f64 = 1e-4; + +// the type III ellipses: their centers, squared semi-axes and angle +const P: [f64; 9] = [0.0, 1.0, 0.0, 1.0, 2.0, 0.0, 1.0, 2.0, 3.0]; +const Q: [f64; 9] = [1.5, 0.5, 2.5, 1.5, 0.5, 3.5, 2.5, 1.5, 0.5]; +const A_SQUARED: f64 = 0.3; +const B_SQUARED: f64 = 1.2; + +// the rays of a two-objective front, and the divisions of a three-objective one's +const RAYS: usize = 20_000; +const DIVISIONS: usize = 288; + +/// A difficulty triplet (η, ζ, γ) of the DAS-CMOP problems, each in [0, 1]: the levels of +/// diversity, feasibility and convergence hardness that the type I, II and III constraints set +/// (see the [`DasCmop1`] docs). +/// +/// ``` +/// use genoxide::multi::problems::Difficulty; +/// +/// let difficulty = Difficulty::standard(8); +/// assert_eq!(difficulty, Difficulty::new(0.5, 0.5, 0.5)); +/// assert_eq!(Difficulty::STANDARD.len(), 16); +/// ``` +#[derive(Clone, Copy, Debug, PartialEq)] +pub struct Difficulty { + diversity: f64, + feasibility: f64, + convergence: f64, +} + +impl Difficulty { + /// The paper's sixteen triplets (table 3), numbered from 1 there: three of one kind of + /// difficulty at 0.25, then at 0.5 and 0.75, each followed by the three at once + /// (1-12), then four with an equality, ζ = 1 (13-16). + pub const STANDARD: [Difficulty; 16] = [ + Self::of(0.25, 0.0, 0.0), + Self::of(0.0, 0.25, 0.0), + Self::of(0.0, 0.0, 0.25), + Self::of(0.25, 0.25, 0.25), + Self::of(0.5, 0.0, 0.0), + Self::of(0.0, 0.5, 0.0), + Self::of(0.0, 0.0, 0.5), + Self::of(0.5, 0.5, 0.5), + Self::of(0.75, 0.0, 0.0), + Self::of(0.0, 0.75, 0.0), + Self::of(0.0, 0.0, 0.75), + Self::of(0.75, 0.75, 0.75), + Self::of(0.0, 1.0, 0.0), + Self::of(0.5, 1.0, 0.0), + Self::of(0.0, 1.0, 0.5), + Self::of(0.5, 1.0, 0.5), + ]; + + const fn of(diversity: f64, feasibility: f64, convergence: f64) -> Self { + Self { + diversity, + feasibility, + convergence, + } + } + + /// The triplet (η, ζ, γ): `diversity` η, `feasibility` ζ and `convergence` γ. + /// + /// # Panics + /// + /// If one isn't in [0, 1]. + pub fn new(diversity: f64, feasibility: f64, convergence: f64) -> Self { + for (name, level) in [ + ("diversity", diversity), + ("feasibility", feasibility), + ("convergence", convergence), + ] { + assert!( + (0.0..=1.0).contains(&level), + "a difficulty's {name} is in [0, 1], not {level}" + ); + } + Self::of(diversity, feasibility, convergence) + } + + /// The paper's triplet `number`, from 1 to 16 (table 3): [`STANDARD`](Self::STANDARD)'s + /// `number − 1`. + /// + /// # Panics + /// + /// If `number` isn't from 1 to 16. + pub fn standard(number: usize) -> Self { + assert!( + (1..=16).contains(&number), + "the paper's difficulty triplets are numbered 1 to 16, not {number}" + ); + Self::STANDARD[number - 1] + } + + /// η, the diversity hardness: the type I constraints' `b = 2η − 1`. + pub fn diversity(&self) -> f64 { + self.diversity + } + + /// ζ, the feasibility hardness: the type II constraint's `e = 0.5 − ln ζ`. + pub fn feasibility(&self) -> f64 { + self.feasibility + } + + /// γ, the convergence hardness: the type III constraints' `r = γ/2`. + pub fn convergence(&self) -> f64 { + self.convergence + } + + fn b(&self) -> f64 { + 2.0 * self.diversity - 1.0 + } + + fn r(&self) -> f64 { + 0.5 * self.convergence + } + + // the type II constraint as g <= 0 + fn band(&self, g: f64) -> f64 { + let zeta = self.feasibility; + if zeta == 0.0 { + -g + } else if zeta == 1.0 { + (g - D).abs() - TOLERANCE + } else { + let e = D - math::ln(zeta); + -((e - g) * (g - D)) + } + } + + // the least and largest g that the type II constraint allows, before rounding + fn distances(&self) -> (f64, f64) { + let zeta = self.feasibility; + if zeta == 0.0 { + (0.0, f64::INFINITY) + } else if zeta == 1.0 { + (D - TOLERANCE, D + TOLERANCE) + } else { + (D, D - math::ln(zeta)) + } + } + + // a key for the cache of fronts + fn bits(&self) -> [u64; 3] { + [self.diversity, self.feasibility, self.convergence].map(f64::to_bits) + } +} + +// the total violation of constraints g(x) <= 0 +fn violation(values: &[f64]) -> f64 { + values.iter().map(|&g| at_most(g, 0.0)).sum() +} + +fn next_up(x: f64) -> f64 { + if x == 0.0 { + f64::from_bits(1) + } else { + f64::from_bits(x.to_bits() + 1) + } +} + +// ---- the distance functions ---------------------------------------------------------------------- + +// DAS-CMOP1-3: Σⱼ₌₂ⁿ (xⱼ − sin(0.5πx₁))² +fn g_sine(x: &[f64]) -> f64 { + let target = math::sin(0.5 * PI * x[0]); + x[1..].iter().map(|xj| (xj - target) * (xj - target)).sum() +} + +// DAS-CMOP4-8: the variables from `first` (0-based) on, (n − first) + Σ ((xⱼ − 0.5)² − +// cos(20π(xⱼ − 0.5))) +fn g_multimodal(x: &[f64], first: usize) -> f64 { + let sum: f64 = x[first..] + .iter() + .map(|xj| (xj - 0.5) * (xj - 0.5) - math::cos(20.0 * PI * (xj - 0.5))) + .sum(); + (x.len() - first) as f64 + sum +} + +// DAS-CMOP9: Σⱼ₌₃ⁿ (xⱼ − cos(0.25 j π (x₁ + x₂)/n))², j 1-based +fn g_linked(x: &[f64]) -> f64 { + let n = x.len() as f64; + let sum = x[0] + x[1]; + x.iter() + .enumerate() + .skip(2) + .map(|(i, &xj)| { + let d = xj - math::cos(0.25 * PI * (i + 1) as f64 * sum / n); + d * d + }) + .sum() +} + +// ---- the shapes ---------------------------------------------------------------------------------- + +fn concave(x1: f64) -> [f64; 2] { + [x1, 1.0 - x1 * x1] +} + +fn convex(x1: f64) -> [f64; 2] { + [x1, 1.0 - x1.sqrt()] +} + +fn discontinuous(x1: f64) -> [f64; 2] { + [x1, 1.0 - x1.sqrt() + 0.5 * math::sin(5.0 * PI * x1).abs()] +} + +fn simplex(x1: f64, x2: f64) -> [f64; 3] { + [x1 * x2, x2 * (1.0 - x1), 1.0 - x2] +} + +fn sphere(x1: f64, x2: f64) -> [f64; 3] { + let (sin1, cos1) = (math::sin(0.5 * PI * x1), math::cos(0.5 * PI * x1)); + [ + cos1 * math::cos(0.5 * PI * x2), + cos1 * math::sin(0.5 * PI * x2), + sin1, + ] +} + +// ---- the constraints ----------------------------------------------------------------------------- + +// the type I constraint on x₁ as g <= 0 +fn diversity_sine(difficulty: &Difficulty, x1: f64) -> f64 { + difficulty.b() - math::sin(A * PI * x1) +} + +// the type I constraint on x₂ as g <= 0 +fn diversity_cosine(difficulty: &Difficulty, x2: f64) -> f64 { + difficulty.b() - math::cos(A * PI * x2) +} + +// the nine ellipses' forms at f, each feasible at r or more +fn ellipses(f: [f64; 2]) -> [f64; 9] { + let (sin, cos) = (math::sin(-0.25 * PI), math::cos(-0.25 * PI)); + std::array::from_fn(|k| { + let (u, v) = (f[0] - P[k], f[1] - Q[k]); + let first = u * cos - v * sin; + let second = u * sin + v * cos; + first * first / A_SQUARED + second * second / B_SQUARED + }) +} + +// the four spheres' centers +fn centers() -> [[f64; 3]; 4] { + let middle = 1.0 / 3.0f64.sqrt(); + [ + [1.0, 0.0, 0.0], + [0.0, 1.0, 0.0], + [0.0, 0.0, 1.0], + [middle, middle, middle], + ] +} + +// the constraints of a two-objective problem as g <= 0, from x₁, g and f +fn two_values(difficulty: &Difficulty, x1: f64, g: f64, f: [f64; 2]) -> [f64; 11] { + let r = difficulty.r(); + let forms = ellipses(f); + std::array::from_fn(|k| match k { + 0 => diversity_sine(difficulty, x1), + 1 => difficulty.band(g), + _ => r - forms[k - 2], + }) +} + +// the constraints of a three-objective problem as g <= 0, from x₁, x₂, g and f +fn three_values(difficulty: &Difficulty, x1: f64, x2: f64, g: f64, f: [f64; 3]) -> [f64; 7] { + let r = difficulty.r(); + let centers = centers(); + std::array::from_fn(|k| match k { + 0 => diversity_sine(difficulty, x1), + 1 => diversity_cosine(difficulty, x2), + 2 => difficulty.band(g), + _ => { + let c = centers[k - 3]; + let squares: f64 = (0..3).map(|j| (f[j] - c[j]) * (f[j] - c[j])).sum(); + r * r - squares + } + }) +} + +// ---- the fronts ---------------------------------------------------------------------------------- + +// the intervals of g along the ray α + g (1, …, 1) inside an ellipse (two objectives) or a sphere +// (three), open: the boundary is feasible +fn blocked(difficulty: &Difficulty, alpha: [f64; M]) -> Vec<(f64, f64)> { + let r = difficulty.r(); + let mut intervals = Vec::new(); + if r == 0.0 { + return intervals; + } + if M == 2 { + // along the ray, the first rotated coordinate u grows by √2 g and the second is fixed: + // inside where (u₀ + √2 g)² < a² (r − v²/b²) + for k in 0..9 { + let (du, dv) = (alpha[0] - P[k], alpha[1] - Q[k]); + let u = (du + dv) / 2f64.sqrt(); + let v = (dv - du) / 2f64.sqrt(); + let squared = A_SQUARED * (r - v * v / B_SQUARED); + if squared > 0.0 { + let w = squared.sqrt(); + intervals.push(((-w - u) / 2f64.sqrt(), (w - u) / 2f64.sqrt())); + } + } + } else { + // |α − c + g 1|² < r²: 3g² + 2Sg + |α − c|² − r² < 0, S = Σ (αⱼ − cⱼ) + for c in centers() { + let w: [f64; M] = std::array::from_fn(|j| alpha[j] - c[j]); + let sum: f64 = w.iter().sum(); + let squares: f64 = w.iter().map(|v| v * v).sum(); + let discriminant = sum * sum - 3.0 * (squares - r * r); + if discriminant > 0.0 { + let root = discriminant.sqrt(); + intervals.push(((-sum - root) / 3.0, (-sum + root) / 3.0)); + } + } + } + intervals +} + +// the first feasible g on the ray α + g (1, …, 1), if any: the least g of the type II band, +// moved past every ellipse or sphere it's in, then up by a few floating-point steps where +// rounding leaves a constraint just broken; `feasible` checks the constraints at g +fn first_feasible( + difficulty: &Difficulty, + alpha: [f64; M], + feasible: impl Fn(f64) -> bool, +) -> Option { + let (low, high) = difficulty.distances(); + let intervals = blocked(difficulty, alpha); + let mut g = low; + loop { + let inside = intervals.iter().find(|&&(start, end)| g > start && g < end); + match inside { + Some(&(_, end)) => g = end, + None => break, + } + } + if g > high + 1e-12 * high.abs().max(1.0) { + return None; + } + for _ in 0..1_000 { + if feasible(g) { + return Some(g); + } + g = next_up(g); + } + None +} + +// the intervals of x in [0, 1] where `allowed(x) <= 0`, closed, their ends the last allowed +// floating-point values (found by bisection between `steps` evenly spread values) +fn intervals(allowed: impl Fn(f64) -> f64, steps: usize) -> Vec<(f64, f64)> { + let inside = |v: f64| allowed(v) <= 0.0; + let boundary = |mut a: f64, mut b: f64| { + let side = inside(a); + loop { + let middle = 0.5 * (a + b); + if middle <= a.min(b) || middle >= a.max(b) { + return a; + } + if inside(middle) == side { + a = middle; + } else { + b = middle; + } + } + }; + let mut found = Vec::new(); + let mut start = inside(0.0).then_some(0.0); + let mut previous = 0.0; + for i in 1..=steps { + let v = i as f64 / steps as f64; + match (start, inside(v)) { + (Some(low), false) => { + found.push((low, boundary(previous, v))); + start = None; + } + (None, true) => start = Some(boundary(v, previous)), + _ => {} + } + previous = v; + } + if let Some(low) = start { + found.push((low, 1.0)); + } + found +} + +// the dense front of a two-objective problem with the shape `alpha`: the first feasible point of +// each ray, non-dominated, sorted by f₁ +fn two_front(difficulty: &Difficulty, alpha: fn(f64) -> [f64; 2]) -> Vec<[f64; 2]> { + let allowed = |x1: f64| diversity_sine(difficulty, x1); + let mut rays: Vec = (0..=RAYS) + .map(|i| i as f64 / RAYS as f64) + .filter(|&x1| allowed(x1) <= 0.0) + .collect(); + for (low, high) in intervals(allowed, RAYS) { + rays.extend([low, high]); + } + let points = rays + .into_iter() + .filter_map(|x1| { + let a = alpha(x1); + let feasible = |g: f64| { + let f = [a[0] + g, a[1] + g]; + two_values(difficulty, x1, g, f) + .iter() + .all(|&value| value <= 0.0) + }; + first_feasible(difficulty, a, feasible).map(|g| [a[0] + g, a[1] + g]) + }) + .collect(); + non_dominated(points) +} + +// the first feasible points of the rays of Das and Dennis's points with `divisions`, mapped to +// the position variables by `positions` +fn three_rays( + difficulty: &Difficulty, + alpha: fn(f64, f64) -> [f64; 3], + positions: fn([f64; 3]) -> (f64, f64), + divisions: usize, +) -> Vec<[f64; 3]> { + das_dennis::<3>(divisions) + .into_iter() + .filter_map(|w| { + let (x1, x2) = positions(w); + if diversity_sine(difficulty, x1) > 0.0 || diversity_cosine(difficulty, x2) > 0.0 { + return None; + } + let a = alpha(x1, x2); + let feasible = |g: f64| { + let f = a.map(|v| v + g); + three_values(difficulty, x1, x2, g, f) + .iter() + .all(|&value| value <= 0.0) + }; + first_feasible(difficulty, a, feasible).map(|g| a.map(|v| v + g)) + }) + .collect() +} + +// DAS-CMOP7's position variables for the simplex point w: x₂ = 1 − w₃, x₁ = w₁/x₂ +fn simplex_positions(w: [f64; 3]) -> (f64, f64) { + let x2 = (1.0 - w[2]).clamp(0.0, 1.0); + let x1 = if x2 > 0.0 { + (w[0] / x2).clamp(0.0, 1.0) + } else { + 0.0 + }; + (x1, x2) +} + +// DAS-CMOP8's and DAS-CMOP9's position variables for the direction w, on the unit sphere +fn sphere_positions(w: [f64; 3]) -> (f64, f64) { + let norm = w.iter().map(|v| v * v).sum::().sqrt(); + let s = w.map(|v| v / norm); + let x1 = (math::asin(s[2].clamp(0.0, 1.0)) / (0.5 * PI)).clamp(0.0, 1.0); + let x2 = if s[0] > 0.0 || s[1] > 0.0 { + (math::atan2(s[1], s[0]) / (0.5 * PI)).clamp(0.0, 1.0) + } else { + 0.0 + }; + (x1, x2) +} + +// a total order on f64, for the staircase of `non_dominated_3` +#[derive(Clone, Copy, PartialEq)] +struct Ordered(f64); + +impl Eq for Ordered {} + +impl PartialOrd for Ordered { + fn partial_cmp(&self, other: &Self) -> Option { + Some(self.cmp(other)) + } +} + +impl Ord for Ordered { + fn cmp(&self, other: &Self) -> Ordering { + self.0.total_cmp(&other.0) + } +} + +// the points of three objectives that no other point dominates, each once, sorted: in order of +// f₁ (then f₂, f₃), each point is dominated by an earlier one iff the earlier ones' staircase of +// (f₂, f₃) has a point at or below it +fn non_dominated_3(mut points: Vec<[f64; 3]>) -> Vec<[f64; 3]> { + points.sort_by(|a, b| { + a[0].total_cmp(&b[0]) + .then(a[1].total_cmp(&b[1])) + .then(a[2].total_cmp(&b[2])) + }); + // f₂ → f₃, f₃ falling as f₂ rises + let mut staircase: BTreeMap = BTreeMap::new(); + let mut front = Vec::new(); + for p in points { + let below = staircase.range(..=Ordered(p[1])).next_back(); + if below.is_some_and(|(_, &f3)| f3 <= p[2]) { + continue; + } + let covered: Vec = staircase + .range(Ordered(p[1])..) + .take_while(|(_, f3)| **f3 >= p[2]) + .map(|(key, _)| *key) + .collect(); + for key in covered { + staircase.remove(&key); + } + staircase.insert(Ordered(p[1]), p[2]); + front.push(p); + } + front +} + +// the dense fronts, computed once per problem and triplet +type Fronts = Mutex>>>; + +fn cached( + cache: &'static OnceLock>, + problem: u8, + difficulty: &Difficulty, + compute: impl FnOnce() -> Vec<[f64; M]>, +) -> Arc> { + let cache = cache.get_or_init(|| Mutex::new(HashMap::new())); + let key = (problem, difficulty.bits()); + if let Some(front) = cache.lock().expect("not poisoned").get(&key) { + return Arc::clone(front); + } + let front = Arc::new(compute()); + let mut cache = cache.lock().expect("not poisoned"); + Arc::clone(cache.entry(key).or_insert(front)) +} + +// the least and largest of each objective on a front +fn corners(front: &[[f64; M]]) -> ([f64; M], [f64; M]) { + let low = std::array::from_fn(|j| front.iter().map(|f| f[j]).fold(f64::INFINITY, f64::min)); + let high = + std::array::from_fn(|j| front.iter().map(|f| f[j]).fold(f64::NEG_INFINITY, f64::max)); + (low, high) +} + +// at least `points` points of a three-objective front: the rays of the fewest divisions that +// divide 288 (or are multiples of it), so that they're among the dense front's, with the dense +// front's points where an objective is least or largest +fn three_front( + dense: &[[f64; 3]], + points: usize, + rays: impl Fn(usize) -> Vec<[f64; 3]>, +) -> Vec<[f64; 3]> { + if points == 0 || dense.is_empty() { + return Vec::new(); + } + let (low, high) = corners(dense); + let extremes: Vec<[f64; 3]> = (0..3) + .flat_map(|j| { + let least = dense.iter().find(|f| f[j] == low[j]); + let largest = dense.iter().find(|f| f[j] == high[j]); + [least, largest] + }) + .flatten() + .copied() + .collect(); + let least = divisions_for::<3>(points); + let candidates = (1..=DIVISIONS) + .filter(|d| DIVISIONS.is_multiple_of(*d)) + .chain((2..).map(|m| m * DIVISIONS)); + for divisions in candidates.filter(|&d| d >= least) { + let mut front = rays(divisions); + front.extend(extremes.iter().copied()); + let front = non_dominated_3(front); + if front.len() >= points { + return front; + } + } + unreachable!("the candidates go on") +} + +// ---- the problems -------------------------------------------------------------------------------- + +// the struct, sizes, fitness and metadata of a DAS-CMOP problem +macro_rules! das_cmop { + ( + $(#[$doc:meta])* + $name:ident, $label:literal, $number:literal, $m:literal, $count:literal, + ($diversity:literal, $feasibility:literal, $convergence:literal) + ) => { + $(#[$doc])* + #[derive(Clone, Copy, Debug, PartialEq)] + pub struct $name { + variables: usize, + difficulty: Difficulty, + } + + impl $name { + /// The number of constraints. + pub const CONSTRAINTS: usize = $count; + + /// The problem with `variables` variables and the difficulty triplet `difficulty`. + /// + /// # Panics + /// + #[doc = concat!("With fewer than ", stringify!($m), " variables.")] + pub fn new(variables: usize, difficulty: Difficulty) -> Self { + assert!( + variables >= $m, + concat!($label, " needs at least ", stringify!($m), " variables") + ); + Self { + variables, + difficulty, + } + } + + /// The problem with the paper's 30 variables and the difficulty triplet + /// `difficulty`. + pub fn with_difficulty(difficulty: Difficulty) -> Self { + Self::new(VARIABLES, difficulty) + } + + /// The number of variables. + pub fn variables(&self) -> usize { + self.variables + } + + /// The difficulty triplet. + pub fn difficulty(&self) -> Difficulty { + self.difficulty + } + + /// The triplet that the paper plots this problem's front with (figure 6). + pub fn figure_difficulty() -> Difficulty { + Difficulty::of($diversity, $feasibility, $convergence) + } + } + + impl Default for $name { + #[doc = concat!("The problem with the paper's 30 variables and the triplet of its figure 6, (", stringify!($diversity), ", ", stringify!($feasibility), ", ", stringify!($convergence), ").")] + fn default() -> Self { + Self::with_difficulty(Self::figure_difficulty()) + } + } + + impl MultiFitnessFunction for $name { + type Output = ([f64; $m], f64); + + /// The objective values of `x` and its constraint violation, 0 when it's feasible. + /// + /// # Panics + /// + #[doc = concat!("If `x` has fewer than ", stringify!($m), " genes.")] + fn evaluate(&self, x: &Reals) -> ([f64; $m], f64) { + let (f, values) = self.parts(x); + (f, violation(&values)) + } + } + + impl MultiProblem<$m> for $name { + type Representation = Real; + + fn name(&self) -> &'static str { + $label + } + + fn representation(&self) -> Real { + Real::uniform(self.variables, 0.0..=1.0).expect("valid bounds") + } + + fn reference(&self) -> &'static str { + REFERENCE + } + + fn reference_url(&self) -> Option<&'static str> { + Some(REFERENCE_URL) + } + + fn constraint_count(&self) -> usize { + $count + } + + fn constraints(&self, genome: &Reals) -> Constraints { + Constraints::new(self.parts(genome).1.to_vec(), Vec::new()) + } + + fn optimal_front(&self, points: usize) -> Option> { + Some(self.front(points)) + } + + fn ideal_point(&self) -> Option<[f64; $m]> { + Some(corners(&self.dense()).0) + } + + fn nadir_point(&self) -> Option<[f64; $m]> { + Some(corners(&self.dense()).1) + } + } + }; +} + +// the objectives and constraints of a two-objective problem with the shape `$alpha` and the +// distance function `$g`, and its front +macro_rules! two_objectives { + ($name:ident, $number:literal, $alpha:ident, |$x:ident| $g:expr) => { + impl $name { + fn parts(&self, $x: &Reals) -> ([f64; 2], [f64; 11]) { + let g = $g; + let a = $alpha($x[0]); + let f = [a[0] + g, a[1] + g]; + (f, two_values(&self.difficulty, $x[0], g, f)) + } + + fn dense(&self) -> Arc> { + static CACHE: OnceLock> = OnceLock::new(); + cached(&CACHE, $number, &self.difficulty, || { + two_front(&self.difficulty, $alpha) + }) + } + + fn front(&self, points: usize) -> Vec<[f64; 2]> { + let dense = self.dense(); + if dense.is_empty() { + return Vec::new(); + } + spread_by_length(&dense, points) + } + } + }; +} + +// the same for three objectives, with the position variables of a direction `$positions` +macro_rules! three_objectives { + ($name:ident, $number:literal, $alpha:ident, $positions:ident, |$x:ident| $g:expr) => { + impl $name { + fn parts(&self, $x: &Reals) -> ([f64; 3], [f64; 7]) { + let g = $g; + let a = $alpha($x[0], $x[1]); + let f = a.map(|v| v + g); + (f, three_values(&self.difficulty, $x[0], $x[1], g, f)) + } + + fn rays(&self, divisions: usize) -> Vec<[f64; 3]> { + three_rays(&self.difficulty, $alpha, $positions, divisions) + } + + fn dense(&self) -> Arc> { + static CACHE: OnceLock> = OnceLock::new(); + cached(&CACHE, $number, &self.difficulty, || { + non_dominated_3(self.rays(DIVISIONS)) + }) + } + + fn front(&self, points: usize) -> Vec<[f64; 3]> { + three_front(&self.dense(), points, |divisions| self.rays(divisions)) + } + } + }; +} + +das_cmop!( + /// DAS-CMOP1: `f₁ = x₁ + g`, `f₂ = 1 − x₁² + g`, `g = Σⱼ₌₂ⁿ (xⱼ − sin(0.5πx₁))²`, subject to + /// 11 constraints: type I on x₁, type II on g and nine type III ellipses. + /// + /// n variables in [0, 1], 30 by default; a [`Difficulty`] triplet (η, ζ, γ), by default + /// (0, 0.5, 0.5), the paper's figure 6. The unconstrained front is concave, `f₂ = 1 − f₁²` at + /// g = 0. The constraints: + /// + /// - type I: `sin(20πx₁) ≥ 2η − 1`, which cuts x₁ into ten intervals and the front into as + /// many pieces, narrower as η grows (none at η = 0); + /// - type II: `(e − g)(g − 0.5) ≥ 0` with `e = 0.5 − ln ζ`, which asks g between 0.5 and e, + /// moving the front out along the diagonal by 0.5 (any g at ζ = 0, and g = 0.5 within + /// 10⁻⁴ at ζ = 1); + /// - type III: the outside of nine ellipses rotated by 45°, centered at (0, 1.5), (1, 0.5), + /// (0, 2.5), (1, 1.5), (2, 0.5), (0, 3.5), (1, 2.5), (2, 1.5) and (3, 0.5), of squared + /// semi-axes 0.3 r and 1.2 r, r = γ/2, which block the way to the front, and, where they + /// reach it, put pieces of it on their boundaries. + /// + /// The front depends on the triplet: [`optimal_front`](MultiProblem::optimal_front) samples + /// it (see the module docs, and [`DasCmop4`], [`DasCmop7`] for the other distance and + /// shape functions). The distance function here is smooth and easy; the difficulty is all in + /// the constraints. + /// + /// Fan, Z. et al. (2020). Difficulty adjustable and scalable constrained multiobjective + /// test problem toolkit. *Evolutionary Computation* 28(3): 339-378, table 2. + DasCmop1, + "DAS-CMOP1", + 1, + 2, + 11, + (0.0, 0.5, 0.5) +); + +das_cmop!( + /// DAS-CMOP2: `f₁ = x₁ + g`, `f₂ = 1 − √x₁ + g`, with DAS-CMOP1's g and constraints. + /// + /// n variables in [0, 1], 30 by default; a [`Difficulty`] triplet, by default (0, 0.5, 0.5), + /// the paper's figure 6. The unconstrained front is convex, `f₂ = 1 − √f₁`; the constraints + /// are [`DasCmop1`]'s. + /// + /// Fan, Z. et al. (2020). Difficulty adjustable and scalable constrained multiobjective + /// test problem toolkit. *Evolutionary Computation* 28(3): 339-378, table 2. + DasCmop2, + "DAS-CMOP2", + 2, + 2, + 11, + (0.0, 0.5, 0.5) +); + +das_cmop!( + /// DAS-CMOP3: `f₁ = x₁ + g`, `f₂ = 1 − √x₁ + 0.5 |sin(5πx₁)| + g`, with DAS-CMOP1's g and + /// constraints. + /// + /// n variables in [0, 1], 30 by default; a [`Difficulty`] triplet, by default (0, 0.5, 0.5), + /// the paper's figure 6. The unconstrained front is disconnected: the non-dominated parts of + /// a convex curve with five bumps. The constraints are [`DasCmop1`]'s. + /// + /// Fan, Z. et al. (2020). Difficulty adjustable and scalable constrained multiobjective + /// test problem toolkit. *Evolutionary Computation* 28(3): 339-378, table 2. + DasCmop3, + "DAS-CMOP3", + 3, + 2, + 11, + (0.0, 0.5, 0.5) +); + +das_cmop!( + /// DAS-CMOP4: [`DasCmop1`]'s shape and constraints, with the multimodal distance function + /// `g = (n − 1) + Σⱼ₌₂ⁿ ((xⱼ − 0.5)² − cos(20π(xⱼ − 0.5)))`. + /// + /// n variables in [0, 1], 30 by default; a [`Difficulty`] triplet, by default + /// (0.5, 0.5, 0.5), the paper's figure 6. g is 0 with every distance variable at 0.5, and + /// has 11ⁿ⁻¹ − 1 local minima, the cosine's (as DTLZ1's): an algorithm climbs down them, and + /// the type II band (g between 0.5 and e) is one more trap among them. The front is + /// DAS-CMOP1's for the same triplet. + /// + /// Fan, Z. et al. (2020). Difficulty adjustable and scalable constrained multiobjective + /// test problem toolkit. *Evolutionary Computation* 28(3): 339-378, table 2. + DasCmop4, + "DAS-CMOP4", + 4, + 2, + 11, + (0.5, 0.5, 0.5) +); + +das_cmop!( + /// DAS-CMOP5: [`DasCmop2`]'s shape, `f₂ = 1 − √x₁ + g`, with [`DasCmop4`]'s multimodal g and + /// [`DasCmop1`]'s constraints. + /// + /// n variables in [0, 1], 30 by default; a [`Difficulty`] triplet, by default + /// (0.5, 0.5, 0.5), the paper's figure 6. The front is DAS-CMOP2's for the same triplet. + /// + /// Fan, Z. et al. (2020). Difficulty adjustable and scalable constrained multiobjective + /// test problem toolkit. *Evolutionary Computation* 28(3): 339-378, table 2. + DasCmop5, + "DAS-CMOP5", + 5, + 2, + 11, + (0.5, 0.5, 0.5) +); + +das_cmop!( + /// DAS-CMOP6: [`DasCmop3`]'s shape, `f₂ = 1 − √x₁ + 0.5 |sin(5πx₁)| + g`, with + /// [`DasCmop4`]'s multimodal g and [`DasCmop1`]'s constraints. + /// + /// n variables in [0, 1], 30 by default; a [`Difficulty`] triplet, by default + /// (0.5, 0.5, 0.5), the paper's figure 6. The front is DAS-CMOP3's for the same triplet. + /// + /// Fan, Z. et al. (2020). Difficulty adjustable and scalable constrained multiobjective + /// test problem toolkit. *Evolutionary Computation* 28(3): 339-378, table 2. + DasCmop6, + "DAS-CMOP6", + 6, + 2, + 11, + (0.5, 0.5, 0.5) +); + +das_cmop!( + /// DAS-CMOP7: three objectives, `f₁ = x₁x₂ + g`, `f₂ = x₂(1 − x₁) + g`, `f₃ = 1 − x₂ + g`, + /// `g = (n − 2) + Σⱼ₌₃ⁿ ((xⱼ − 0.5)² − cos(20π(xⱼ − 0.5)))`, subject to 7 constraints. + /// + /// n variables in [0, 1], 30 by default; a [`Difficulty`] triplet (η, ζ, γ), by default + /// (0.5, 0.5, 0.5), the paper's figure 6. The unconstrained front is the simplex + /// `f₁ + f₂ + f₃ = 1`. The constraints: + /// + /// - type I: `sin(20πx₁) ≥ 2η − 1` and `cos(20πx₂) ≥ 2η − 1`, which cut the front into a + /// grid of patches; + /// - type II: `(e − g)(g − 0.5) ≥ 0`, as [`DasCmop1`]'s; + /// - type III: the outside of four spheres of radius r = γ/2, centered at the unit vectors, + /// the simplex's corners, and at (1, 1, 1)/√3: `Σⱼ fⱼ² − fₖ² + (fₖ − 1)² ≥ r²` and + /// `Σⱼ (fⱼ − 1/√3)² ≥ r²`. Near the corners they push the front out along the diagonal. + /// + /// Fan, Z. et al. (2020). Difficulty adjustable and scalable constrained multiobjective + /// test problem toolkit. *Evolutionary Computation* 28(3): 339-378, table 2. + DasCmop7, + "DAS-CMOP7", + 7, + 3, + 7, + (0.5, 0.5, 0.5) +); + +das_cmop!( + /// DAS-CMOP8: three objectives on the sphere, `f₁ = cos(0.5πx₁) cos(0.5πx₂) + g`, + /// `f₂ = cos(0.5πx₁) sin(0.5πx₂) + g`, `f₃ = sin(0.5πx₁) + g`, with [`DasCmop7`]'s g and + /// constraints. + /// + /// n variables in [0, 1], 30 by default; a [`Difficulty`] triplet, by default + /// (0.5, 0.5, 0.5), the paper's figure 6. The unconstrained front is the unit sphere's + /// octant, whose middle, (1, 1, 1)/√3, is the fourth type III sphere's center: the front has a + /// hole there. + /// + /// Fan, Z. et al. (2020). Difficulty adjustable and scalable constrained multiobjective + /// test problem toolkit. *Evolutionary Computation* 28(3): 339-378, table 2. + DasCmop8, + "DAS-CMOP8", + 8, + 3, + 7, + (0.5, 0.5, 0.5) +); + +das_cmop!( + /// DAS-CMOP9: [`DasCmop8`]'s shape and [`DasCmop7`]'s constraints, with the distance + /// function `g = Σⱼ₌₃ⁿ (xⱼ − cos(0.25 j π (x₁ + x₂)/n))²`. + /// + /// n variables in [0, 1], 30 by default; a [`Difficulty`] triplet, by default + /// (0.5, 0.5, 0.5), the paper's figure 6. g is 0 where each distance variable is a cosine of + /// the position variables, different for each: the variables are linked, and no single value + /// of xⱼ is right everywhere. The front is DAS-CMOP8's for the same triplet. + /// + /// Fan, Z. et al. (2020). Difficulty adjustable and scalable constrained multiobjective + /// test problem toolkit. *Evolutionary Computation* 28(3): 339-378, table 2. + DasCmop9, + "DAS-CMOP9", + 9, + 3, + 7, + (0.5, 0.5, 0.5) +); + +two_objectives!(DasCmop1, 1, concave, |x| g_sine(x)); +two_objectives!(DasCmop2, 2, convex, |x| g_sine(x)); +two_objectives!(DasCmop3, 3, discontinuous, |x| g_sine(x)); +two_objectives!(DasCmop4, 4, concave, |x| g_multimodal(x, 1)); +two_objectives!(DasCmop5, 5, convex, |x| g_multimodal(x, 1)); +two_objectives!(DasCmop6, 6, discontinuous, |x| g_multimodal(x, 1)); +three_objectives!(DasCmop7, 7, simplex, simplex_positions, |x| g_multimodal( + x, 2 +)); +three_objectives!(DasCmop8, 8, sphere, sphere_positions, |x| g_multimodal( + x, 2 +)); +three_objectives!(DasCmop9, 9, sphere, sphere_positions, |x| g_linked(x)); + +#[cfg(test)] +mod tests { + use super::*; + use crate::Objective::Minimize; + use crate::StreamRng; + use crate::genome::Representation; + use crate::multi::{Scores, non_dominated_sort}; + + fn at(values: &[f64]) -> Reals { + Reals::from(values.to_vec()) + } + + fn assert_close(actual: &[f64], expected: &[f64]) { + assert_eq!(actual.len(), expected.len()); + for (a, e) in actual.iter().zip(expected) { + assert!( + (a - e).abs() <= 1e-12 * e.abs().max(1.0), + "{actual:?} is not {expected:?}" + ); + } + } + + #[test] + fn the_papers_triplets() { + assert_eq!(Difficulty::standard(1), Difficulty::new(0.25, 0.0, 0.0)); + assert_eq!(Difficulty::standard(12), Difficulty::new(0.75, 0.75, 0.75)); + assert_eq!(Difficulty::standard(16), Difficulty::new(0.5, 1.0, 0.5)); + // 1-12: three of one kind at a level, then all three at it + for level in 0..3 { + let value = 0.25 * (level + 1) as f64; + let four = &Difficulty::STANDARD[4 * level..4 * level + 4]; + assert_eq!(four[0], Difficulty::new(value, 0.0, 0.0)); + assert_eq!(four[1], Difficulty::new(0.0, value, 0.0)); + assert_eq!(four[2], Difficulty::new(0.0, 0.0, value)); + assert_eq!(four[3], Difficulty::new(value, value, value)); + } + // 13-16: ζ = 1 + assert!( + Difficulty::STANDARD[12..] + .iter() + .all(|d| d.feasibility() == 1.0) + ); + assert_eq!( + DasCmop1::default().difficulty(), + Difficulty::new(0.0, 0.5, 0.5) + ); + assert_eq!(DasCmop4::default().difficulty(), Difficulty::standard(8)); + assert_eq!(DasCmop9::default().variables(), 30); + } + + #[test] + #[should_panic(expected = "in [0, 1], not 1.5")] + fn a_level_above_1_panics() { + let _ = Difficulty::new(0.0, 1.5, 0.0); + } + + // the formulas of the paper's table 2 and its code, at points chosen so that the values + // follow by hand + #[test] + fn values_match_the_paper() { + // DAS-CMOP1, (0, 0.5, 0.5): x₁ = 0.5, the distance variables at sin(π/4) but the last, + // 0.5 away: g = 0.25, f = (0.75, 1); type I off (b = −1: −1 − sin 10π); the band + // asks g in [0.5, 0.5 + ln 2], broken by −(e − 0.25)(0.25 − 0.5); the ellipse at + // (1, 1.5): u = (−0.25 − 0.5)/√2, v = (−0.5 + 0.25)/√2, form 0.75²/(2 · 0.3) + 0.25²/ + // (2 · 1.2), r = 0.25 − that + let problem = DasCmop1::default(); + let s = math::sin(0.25 * PI); + let mut genes = vec![s; 30]; + genes[0] = 0.5; + genes[29] = s - 0.5; + let x = at(&genes); + let (f, violation) = problem.evaluate(&x); + assert_close(&f, &[0.75, 1.0]); + let values = problem.constraints(&x).inequalities().to_vec(); + assert_eq!(values.len(), 11); + assert_close(&values[..1], &[-1.0 - math::sin(10.0 * PI)]); + let e = 0.5 + math::ln(2.0); + assert_close(&values[1..2], &[(e - 0.25) * 0.25]); + let form = 0.75 * 0.75 / 0.6 + 0.25 * 0.25 / 2.4; + assert_close(&values[5..6], &[0.25 - form]); + let expected: f64 = values.iter().map(|&v| v.max(0.0)).sum(); + assert_close(&[violation], &[expected]); + + // DAS-CMOP2 and DAS-CMOP3 at x₁ = 0.25 on the unconstrained front: f₂ = 0.5, and + // 0.5 + 0.5 |sin(1.25π)| + let mut genes = vec![math::sin(0.125 * PI); 30]; + genes[0] = 0.25; + let x = at(&genes); + assert_close(&DasCmop2::default().evaluate(&x).0, &[0.25, 0.5]); + let bump = 0.5 * math::sin(1.25 * PI).abs(); + assert_close(&DasCmop3::default().evaluate(&x).0, &[0.25, 0.5 + bump]); + + // DAS-CMOP4: g = 0 at 0.5; a distance variable at 0.6 adds 0.01 − cos 2π + 1 = 0.01 + let mut genes = vec![0.5; 30]; + genes[0] = 0.25; + assert_close( + &DasCmop4::default().evaluate(&at(&genes)).0, + &[0.25, 0.9375], + ); + genes[7] = 0.6; + assert_close(&DasCmop5::default().evaluate(&at(&genes)).0, &[0.26, 0.51]); + + // DAS-CMOP7: x₁ = 0.25, x₂ = 0.5, g = 0: (0.125, 0.375, 0.5), on the simplex; at + // (0.5, 0.5, 0.5): type I sin 5π = 0 ≥ 0 and cos 10π = 1 ≥ 0, the band broken, + // the middle sphere's distance² 3 (1/√3 − 0.5)² + let problem = DasCmop7::default(); + let mut genes = vec![0.5; 30]; + genes[0] = 0.25; + let x = at(&genes); + assert_close(&problem.evaluate(&x).0, &[0.125, 0.375, 0.5]); + let values = problem.constraints(&x).inequalities().to_vec(); + assert_eq!(values.len(), 7); + assert_close(&values[..2], &[-math::sin(5.0 * PI), -math::cos(10.0 * PI)]); + assert_close(&values[2..3], &[(0.5 + math::ln(2.0)) * 0.5]); + let middle = 1.0 / 3.0f64.sqrt(); + let squares = (0.125 - middle).powi(2) + (0.375 - middle).powi(2) + (0.5 - middle).powi(2); + assert_close(&values[6..], &[0.0625 - squares]); + + // DAS-CMOP8 and DAS-CMOP9 at x₁ = 1/3, x₂ = 1/2: (√3/4 · √2 · …) on the sphere + let mut genes = vec![0.5; 30]; + genes[0] = 1.0 / 3.0; + let x = at(&genes); + let (f, _) = DasCmop8::default().evaluate(&x); + let c = math::cos(PI / 6.0); + assert_close( + &f, + &[c * math::cos(0.25 * PI), c * math::sin(0.25 * PI), 0.5], + ); + // DAS-CMOP9's distance variables at their optimal values cos(0.25 j π (x₁ + x₂)/n) + for (i, gene) in genes.iter_mut().enumerate().skip(2) { + *gene = math::cos(0.25 * PI * (i + 1) as f64 * (1.0 / 3.0 + 0.5) / 30.0); + } + let (f9, _) = DasCmop9::default().evaluate(&at(&genes)); + assert_close(&f9, &f); + } + + // the type II band at its levels: any g at ζ = 0, [0.5, e] between, 0.5 within 10⁻⁴ at 1 + #[test] + fn the_band_of_distances() { + let band = |zeta: f64, g: f64| Difficulty::new(0.0, zeta, 0.0).band(g); + assert!(band(0.0, 0.0) <= 0.0 && band(0.0, 7.0) <= 0.0); + let e = 0.5 - math::ln(0.25); + assert!(band(0.25, 0.49) > 0.0 && band(0.25, 0.5) <= 0.0); + assert!(band(0.25, e - 1e-9) <= 0.0 && band(0.25, e + 1e-9) > 0.0); + assert!(band(1.0, 0.5) <= 0.0 && band(1.0, 0.50005) <= 0.0 && band(1.0, 0.49995) <= 0.0); + assert!(band(1.0, 0.5002) > 0.0 && band(1.0, 0.4998) > 0.0); + } + + // a genome of DAS-CMOP1 or DAS-CMOP4 at x₁ with the distance g, spread over the distance + // variables (for DAS-CMOP4, within the cosine's central well) + fn genome_1(x1: f64, g: f64) -> Reals { + let target = math::sin(0.5 * PI * x1); + let step = (g / 29.0).sqrt(); + let sign = if target + step <= 1.0 { 1.0 } else { -1.0 }; + let mut genes = vec![target + sign * step; 30]; + genes[0] = x1; + Reals::from(genes) + } + + // the fronts: feasible where they lie (the ray's point at its g), reached by genomes, + // mutually non-dominated, between the ideal and nadir points that they reach + fn check_two(problem: &impl MultiProblem<2, Representation = Real>, points: usize) { + let front = problem.optimal_front(points).expect("known"); + assert_eq!(front.len(), points, "{}", problem.name()); + let scores: Vec> = front.iter().map(|f| Scores::new(*f)).collect(); + assert_eq!(non_dominated_sort(&scores, &[Minimize; 2]).len(), 1); + let (ideal, nadir) = ( + problem.ideal_point().unwrap(), + problem.nadir_point().unwrap(), + ); + assert_eq!(front[0][0], ideal[0]); + assert_eq!(front[points - 1][1], ideal[1]); + assert_eq!(front[points - 1][0], nadir[0]); + assert_eq!(front[0][1], nadir[1]); + } + + #[test] + fn two_objective_fronts() { + for difficulty in Difficulty::STANDARD { + check_two(&DasCmop1::with_difficulty(difficulty), 50); + check_two(&DasCmop2::with_difficulty(difficulty), 50); + check_two(&DasCmop3::with_difficulty(difficulty), 50); + } + // the same fronts for the same shapes + let difficulty = Difficulty::standard(8); + assert_eq!( + DasCmop1::with_difficulty(difficulty).optimal_front(100), + DasCmop4::with_difficulty(difficulty).optimal_front(100) + ); + // with no constraint, the unconstrained front: f₂ = 1 − f₁², f₁ in [0, 1] + let problem = DasCmop1::with_difficulty(Difficulty::new(0.0, 0.0, 0.0)); + let front = problem.optimal_front(101).unwrap(); + assert!( + front + .iter() + .all(|f| (f[1] - (1.0 - f[0] * f[0])).abs() < 1e-12) + ); + assert_eq!((front[0], front[100]), ([0.0, 1.0], [1.0, 0.0])); + // the type II band moves it by 0.5 + let problem = DasCmop1::with_difficulty(Difficulty::new(0.0, 0.5, 0.0)); + let front = problem.optimal_front(101).unwrap(); + assert!(front.iter().all(|f| { + let x1 = f[0] - 0.5; + (f[1] - 0.5 - (1.0 - x1 * x1)).abs() < 1e-12 + })); + // DAS-CMOP1's default: the front reached by genomes with their g + let problem = DasCmop1::default(); + for f in problem.optimal_front(200).unwrap() { + // f₁ − f₂ = x₁ + x₁² − 1 gives x₁, and g = f₁ − x₁ + let x1 = (-1.0 + (1.0 + 4.0 * (f[0] - f[1] + 1.0)).sqrt()) / 2.0; + let (found, violation) = problem.evaluate(&genome_1(x1, f[0] - x1)); + assert!((found[0] - f[0]).abs() < 1e-9 && (found[1] - f[1]).abs() < 1e-9); + assert!(violation < 1e-9, "{f:?}: {violation}"); + } + } + + // a two-objective shape, and the front of a problem with it for a triplet + type Shape = fn(f64) -> [f64; 2]; + type Front = fn(Difficulty) -> Vec<[f64; 2]>; + + // no feasible point is better than the front: random x₁ and g, every feasible point weakly + // dominated by a point of the front, up to its spacing, and dominating none + #[test] + fn random_points_agree_with_the_two_objective_fronts() { + let mut rng = StreamRng::seed_from_u64(9); + let unit = Real::uniform(2, 0.0..=1.0).expect("valid bounds"); + let shapes: [(Shape, Front); 3] = [ + (concave, |d| { + DasCmop1::with_difficulty(d).optimal_front(4_000).unwrap() + }), + (convex, |d| { + DasCmop2::with_difficulty(d).optimal_front(4_000).unwrap() + }), + (discontinuous, |d| { + DasCmop3::with_difficulty(d).optimal_front(4_000).unwrap() + }), + ]; + for difficulty in [3, 4, 8, 12].map(Difficulty::standard) { + for (alpha, front) in shapes { + let front = front(difficulty); + let mut feasible = 0; + for _ in 0..4_000 { + let genome = unit.random_genome(&mut rng); + let (x1, g) = (genome[0], 0.4 + 2.0 * genome[1]); + let a = alpha(x1); + let f = [a[0] + g, a[1] + g]; + if two_values(&difficulty, x1, g, f).iter().any(|&v| v > 0.0) { + continue; + } + feasible += 1; + assert!( + front + .iter() + .any(|q| q[0] <= f[0] + 0.02 && q[1] <= f[1] + 0.02), + "{difficulty:?}: {f:?}" + ); + for q in &front { + assert!(!(f[0] < q[0] - 1e-9 && f[1] < q[1] - 1e-9), "{f:?} {q:?}"); + } + } + assert!(feasible > 20, "{difficulty:?}: {feasible}"); + } + } + } + + #[test] + fn three_objective_fronts() { + for number in [1, 3, 4, 8, 11, 12, 16] { + let difficulty = Difficulty::standard(number); + for problem in [ + &DasCmop7::with_difficulty(difficulty) as &dyn Check, + &DasCmop8::with_difficulty(difficulty), + ] { + problem.check(91); + } + } + // with no constraint: the simplex and the sphere + let none = Difficulty::new(0.0, 0.0, 0.0); + let front = DasCmop7::with_difficulty(none).optimal_front(91).unwrap(); + assert!( + front + .iter() + .all(|f| (f.iter().sum::() - 1.0).abs() < 1e-12) + ); + let front = DasCmop9::with_difficulty(none).optimal_front(91).unwrap(); + assert!( + front + .iter() + .all(|f| (f.iter().map(|v| v * v).sum::() - 1.0).abs() < 1e-12) + ); + // DAS-CMOP8's and DAS-CMOP9's are the same + let difficulty = Difficulty::standard(8); + assert_eq!( + DasCmop8::with_difficulty(difficulty).optimal_front(200), + DasCmop9::with_difficulty(difficulty).optimal_front(200) + ); + // the type III spheres push the corners out: at γ = 0.5, f₁ reaches past 1 near (1, 0, 0) + let problem = DasCmop8::with_difficulty(Difficulty::new(0.0, 0.0, 0.5)); + let nadir = problem.nadir_point().unwrap(); + assert!(nadir[0] > 1.1, "{nadir:?}"); + } + + trait Check { + fn check(&self, points: usize); + } + + impl> Check for P { + fn check(&self, points: usize) { + let front = self.optimal_front(points).expect("known"); + assert!(front.len() >= points, "{}", self.name()); + let scores: Vec> = front.iter().map(|f| Scores::new(*f)).collect(); + assert_eq!(non_dominated_sort(&scores, &[Minimize; 3]).len(), 1); + let (ideal, nadir) = (self.ideal_point().unwrap(), self.nadir_point().unwrap()); + for j in 0..3 { + assert!(front.iter().any(|f| f[j] == ideal[j]), "{}", self.name()); + assert!(front.iter().any(|f| f[j] == nadir[j]), "{}", self.name()); + } + } + } + + // random points for three objectives, as for two: random x₁, x₂ and g + #[test] + fn random_points_agree_with_the_three_objective_fronts() { + let mut rng = StreamRng::seed_from_u64(10); + let unit = Real::uniform(3, 0.0..=1.0).expect("valid bounds"); + for number in [3, 8, 11] { + let difficulty = Difficulty::standard(number); + for (alpha, front) in [ + ( + simplex as fn(f64, f64) -> [f64; 3], + DasCmop7::with_difficulty(difficulty) + .optimal_front(3_000) + .unwrap(), + ), + ( + sphere, + DasCmop8::with_difficulty(difficulty) + .optimal_front(3_000) + .unwrap(), + ), + ] { + let mut feasible = 0; + for _ in 0..3_000 { + let genome = unit.random_genome(&mut rng); + let (x1, x2, g) = (genome[0], genome[1], 1.5 * genome[2]); + let f = alpha(x1, x2).map(|v| v + g); + if three_values(&difficulty, x1, x2, g, f) + .iter() + .any(|&v| v > 0.0) + { + continue; + } + feasible += 1; + assert!( + front.iter().any(|q| (0..3).all(|j| q[j] <= f[j] + 0.05)), + "{number}: {f:?}" + ); + for q in &front { + assert!(!(0..3).all(|j| f[j] < q[j] - 1e-9), "{f:?} {q:?}"); + } + } + assert!(feasible > 20, "{number}: {feasible}"); + } + } + } + + #[test] + fn three_objective_non_domination() { + let points = vec![ + [1.0, 1.0, 1.0], + [0.0, 2.0, 2.0], + [1.0, 1.0, 1.0], + [2.0, 0.0, 2.0], + [1.0, 2.0, 1.0], + [2.0, 2.0, 0.0], + [0.5, 3.0, 3.0], + ]; + assert_eq!( + non_dominated_3(points), + [ + [0.0, 2.0, 2.0], + [1.0, 1.0, 1.0], + [2.0, 0.0, 2.0], + [2.0, 2.0, 0.0] + ] + ); + } + + #[test] + fn sizes() { + assert_eq!(DasCmop1::default().constraint_count(), 11); + assert_eq!(DasCmop9::default().constraint_count(), 7); + assert_eq!( + DasCmop7::new(3, Difficulty::standard(1)) + .representation() + .bounds() + .len(), + 3 + ); + } +} diff --git a/src/multi/problems/dcdtlz.rs b/src/multi/problems/dcdtlz.rs new file mode 100644 index 00000000..79b7a746 --- /dev/null +++ b/src/multi/problems/dcdtlz.rs @@ -0,0 +1,990 @@ +//! DC1-DTLZ1, DC1-DTLZ3, DC2-DTLZ1, DC2-DTLZ3, DC3-DTLZ1 and DC3-DTLZ3: Li, Chen, Fu and Yao's +//! DTLZ problems with constraints on the decision variables, any number of objectives. +//! +//! Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for +//! constrained multiobjective optimization. *IEEE Transactions on Evolutionary Computation* +//! 23(2): 303-315, which names the problems and runs them, and its supplementary document (section +//! 1.2, eqs. 12-16, figures 4-7 and table 2), which defines them; read in the authors' copies, the +//! paper as arXiv:1711.07907 and the supplement as hosted by the first author +//! (). +//! +//! The objectives are C1-DTLZ1's and C1-DTLZ3's, DTLZ1's and DTLZ3's: genoxide's [`Dtlz1`] and +//! [`Dtlz3`]. The constraints act on the decision variables, where C-DTLZ's act on the +//! objectives: type 1 (DC1) on the first position variable, `cos(aπx₁) > b`, which cuts the front +//! into cones from the origin; type 2 (DC2) on the distance function g, `cos(aπg) > b` and +//! `e^(−g) > b`, which keep the front whole and make nearly all of the rest of the space +//! infeasible, with local optima of the violation along the way; type 3 (DC3) on every position +//! variable and on g, which cuts the front into patches and adds DC2's local optima. +//! +//! **What the paper leaves open, and how genoxide settles it.** The supplement gives a = 3 and +//! b = 0.5 for DC1 and no values for DC2 and DC3, and no number of variables. The authors' lab +//! publishes a C++ platform, EMOC (, read only, to +//! compare: it has no license), whose `dcdtlz.cpp` uses a = 3 throughout, b = 0.5 but for +//! DC2-DTLZ1's 0.9: genoxide takes those values, which [`with_parameters`](Dc1Dtlz1::with_parameters) +//! changes. The supplement writes DC3's position constraints for j = 1, …, m, which would include +//! the first distance variable, 0.5 on the front, where `cos(1.5π) = 0` breaks the constraint and +//! makes the whole front infeasible, against its text and figure 7; EMOC constrains the m − 1 +//! position variables, and so does genoxide, which the supplement's table 2 confirms: DC3's +//! feasible share of random solutions, 3.70%, 0.41% and 0.02% for 3, 5 and 8 objectives, is 3⁻ᵐ, +//! a third for each of the m − 1 position constraints and for g's. EMOC computes DTLZ3's g with +//! the factor 10 in place of DTLZ3's 100 (in its C1-DTLZ3 too); genoxide keeps DTLZ3's, as the +//! supplement says that the objectives are C1-DTLZ3's. The number of variables is DTLZ's usual, +//! `M + 4` for DTLZ1 and `M + 9` for DTLZ3 (as C1-DTLZ1 and C1-DTLZ3 have), an assumption. The +//! supplement's table 2 gives DC1 a feasible share of about 10.1%, which is `arccos(b)/π` for +//! b = 0.95, where its text, its figure 4 and EMOC (and its sampled fronts) have b = 0.5, a share +//! of 1/3: genoxide follows the text. The supplement prints the constraints strict ("> b"); a +//! solution here is feasible on the boundary too. +//! +//! **The fronts**, derived here: g = 0 is feasible under every constraint (`cos 0 = 1` and +//! `e⁰ = 1` are above b < 1), and the position constraints don't involve g, so DTLZ's front +//! points are optimal wherever their position variables are feasible, and no other point is (one +//! with g > 0 is dominated by the same position variables at g = 0). DC2's front is DTLZ's, whole; +//! DC1's and DC3's are the parts of it with feasible position variables. Their +//! [`optimal_front`](MultiProblem::optimal_front) maps Das and Dennis's points to the front (the +//! plane `Σ fᵢ = 1/2` or the unit sphere), finds their position variables, keeps the feasible +//! ones, and adds the points where each variable is at the ends of its feasible values (its least, +//! 0, and its largest), where each objective is least and largest; for 2 objectives, it spreads +//! exactly the requested points by length along a dense sample of feasible x₁. Every point is +//! evaluated from a genome, the position variables with the distance variables at 0.5. They agree +//! with the sampled fronts that EMOC ships for 2 and 3 objectives (`pf_data/dc*dtlz`, compared +//! only): the mean distance from each point of either to the nearest of the other is at most +//! 0.011. + +use super::dtlz::{rastrigin_g, simplex_points, spherical_front}; +use super::mw::spread_by_length; +use super::{Dtlz1, Dtlz3, DynMultiProblem, MultiProblem, boxed, das_dennis, divisions_for}; +use crate::constraint::at_most; +use crate::genome::{Real, Reals}; +use crate::math; +use crate::multi::MultiFitnessFunction; +use crate::problems::Constraints; +use std::f64::consts::PI; + +const REFERENCE: &str = "Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary \ + algorithm for constrained multiobjective optimization. IEEE Transactions \ + on Evolutionary Computation 23(2): 303-315."; +const REFERENCE_URL: &str = "https://doi.org/10.1109/TEVC.2018.2855411"; + +// the samples of x₁ for a two-objective front, and of the feasible values of a variable +const SAMPLES: usize = 100_000; + +// the total violation of constraints g(x) <= 0 +fn violation(values: &[f64]) -> f64 { + values.iter().map(|&g| at_most(g, 0.0)).sum() +} + +// which of DTLZ1 and DTLZ3 a problem builds on +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +enum Shape { + // DTLZ1: the front is the plane Σ f = 1/2 + Plane, + // DTLZ3: the front is the unit sphere + Sphere, +} + +// which constraints: on x₁ (type 1), on g (type 2), or on every position variable and on g +// (type 3) +#[derive(Clone, Copy, Debug, PartialEq, Eq)] +enum Kind { + One, + Two, + Three, +} + +// what the six problems share: the base problem's shape, the constraints' type, the number of +// variables and the parameters a and b +#[derive(Clone, Copy, Debug, PartialEq)] +struct Core { + shape: Shape, + kind: Kind, + variables: usize, + a: f64, + b: f64, +} + +impl Core { + fn new(label: &str, shape: Shape, kind: Kind, variables: usize, a: f64, b: f64) -> Self { + assert!(M >= 2, "{label} needs at least 2 objectives"); + assert!( + variables >= M, + "{label} needs at least as many variables as objectives" + ); + assert!(a.is_finite() && a > 0.0, "{label} needs a finite a above 0"); + assert!( + b.is_finite() && b > -1.0 && b < 1.0, + "{label} needs a finite b between -1 and 1" + ); + Self { + shape, + kind, + variables, + a, + b, + } + } + + fn objectives(&self, x: &Reals) -> [f64; M] { + match self.shape { + Shape::Plane => Dtlz1::::new(self.variables).evaluate(x), + Shape::Sphere => Dtlz3::::new(self.variables).evaluate(x), + } + } + + // the constraint on one value (a position variable or g): b − cos(aπv) <= 0 + fn cosine(&self, value: f64) -> f64 { + self.b - math::cos(self.a * PI * value) + } + + fn values(&self, x: &Reals) -> Vec { + match self.kind { + Kind::One => vec![self.cosine(x[0])], + Kind::Two => { + let g = rastrigin_g(&x[M - 1..]); + vec![self.cosine(g), self.b - math::exp(-g)] + } + Kind::Three => { + let g = rastrigin_g(&x[M - 1..]); + let mut values: Vec = x[..M - 1].iter().map(|&xi| self.cosine(xi)).collect(); + values.push(self.cosine(g)); + values + } + } + } + + fn count(&self) -> usize { + match self.kind { + Kind::One => 1, + Kind::Two => 2, + Kind::Three => M, + } + } + + fn evaluate(&self, x: &Reals) -> ([f64; M], f64) { + (self.objectives(x), violation(&self.values(x))) + } + + // whether the position variable at `index` (0-based) may take `value` on the front + fn allowed(&self, index: usize, value: f64) -> bool { + match self.kind { + Kind::One => index > 0 || self.cosine(value) <= 0.0, + Kind::Two => true, + Kind::Three => self.cosine(value) <= 0.0, + } + } + + // the point of the front at the position variables `positions`: evaluated from the genome + // with the distance variables at 0.5, where g = 0 + fn at(&self, positions: &[f64]) -> [f64; M] { + let mut genes = positions.to_vec(); + genes.resize(self.variables, 0.5); + self.objectives(&Reals::from(genes)) + } + + // the feasible values of a constrained position variable in [0, 1], as closed intervals whose + // ends are the last feasible floating-point values, found by bisection + fn intervals(&self) -> Vec<(f64, f64)> { + let feasible = |v: f64| self.cosine(v) <= 0.0; + // the boundary between a and b, where feasible(a) != feasible(b): the feasible side's + // last value + let boundary = |mut a: f64, mut b: f64| { + let inside = feasible(a); + loop { + let middle = 0.5 * (a + b); + if middle <= a.min(b) || middle >= a.max(b) { + return a; + } + if feasible(middle) == inside { + a = middle; + } else { + b = middle; + } + } + }; + let mut intervals = Vec::new(); + let mut start = feasible(0.0).then_some(0.0); + let mut previous = 0.0; + for i in 1..=SAMPLES { + let v = i as f64 / SAMPLES as f64; + match (start, feasible(v)) { + (Some(low), false) => { + intervals.push((low, boundary(previous, v))); + start = None; + } + (None, true) => start = Some(boundary(v, previous)), + _ => {} + } + previous = v; + } + if let Some(low) = start { + intervals.push((low, 1.0)); + } + intervals + } + + // the position variables of the front point f: inverting DTLZ1's or DTLZ3's position + // functions, with a variable whose value doesn't matter (a product before it is 0) at 0 + fn positions(&self, f: &[f64; M]) -> Vec { + let mut positions = Vec::with_capacity(M - 1); + match self.shape { + Shape::Plane => { + // f = w/2 with Σ w = 1: f_M = (1 − x₁)/2, f_{M−k+1} = x₁ … x_{k−1} (1 − x_k)/2 + let mut product = 1.0; + for k in 0..M - 1 { + let share = 2.0 * f[M - 1 - k]; + let x = if product > 0.0 { + (1.0 - share / product).clamp(0.0, 1.0) + } else { + 0.0 + }; + positions.push(x); + product *= x; + } + } + Shape::Sphere => { + // f_M = sin θ₁, f_{M−k+1} = cos θ₁ … cos θ_{k−1} sin θ_k, θ = πx/2 + let mut product = 1.0; + for k in 0..M - 1 { + let x = if product > 0.0 { + let ratio = (f[M - 1 - k] / product).clamp(0.0, 1.0); + (math::asin(ratio) / (PI / 2.0)).clamp(0.0, 1.0) + } else { + 0.0 + }; + positions.push(x); + product *= math::cos(PI / 2.0 * x); + } + } + } + positions + } + + // the least and largest value each position variable may take on the front + fn ends(&self) -> Vec<[f64; 2]> { + let constrained = match self.intervals().as_slice() { + [] => [0.0, 0.0], + intervals => [intervals[0].0, intervals[intervals.len() - 1].1], + }; + (0..M - 1) + .map(|index| match (self.kind, index) { + (Kind::Two, _) | (Kind::One, 1..) => [0.0, 1.0], + _ => constrained, + }) + .collect() + } + + // the front points with every position variable at one of its ends: the least and largest + // value of each objective on the front are among them, each objective being a product of + // factors monotonic in separate variables + fn corners(&self) -> Vec<[f64; M]> { + let ends = self.ends(); + (0..1usize << (M - 1)) + .map(|choice| { + let positions: Vec = (0..M - 1).map(|i| ends[i][(choice >> i) & 1]).collect(); + self.at(&positions) + }) + .collect() + } + + fn ideal(&self) -> [f64; M] { + if self.kind == Kind::Two { + return [0.0; M]; + } + let corners = self.corners(); + std::array::from_fn(|j| corners.iter().map(|f| f[j]).fold(f64::INFINITY, f64::min)) + } + + fn nadir(&self) -> [f64; M] { + if self.kind == Kind::Two { + return match self.shape { + Shape::Plane => [0.5; M], + Shape::Sphere => [1.0; M], + }; + } + let corners = self.corners(); + std::array::from_fn(|j| { + corners + .iter() + .map(|f| f[j]) + .fold(f64::NEG_INFINITY, f64::max) + }) + } + + fn front(&self, points: usize) -> Vec<[f64; M]> { + if points == 0 { + return Vec::new(); + } + if self.kind == Kind::Two { + return match self.shape { + Shape::Plane => simplex_points::(points) + .into_iter() + .map(|w| w.map(|v| v / 2.0)) + .collect(), + Shape::Sphere => spherical_front::(points), + }; + } + if M == 2 { + return self.two_objectives(points); + } + let corners = self.corners(); + let mut divisions = divisions_for::(points); + loop { + let mut front: Vec<[f64; M]> = das_dennis::(divisions) + .into_iter() + .filter_map(|w| { + let f = match self.shape { + Shape::Plane => w.map(|v| v / 2.0), + Shape::Sphere => { + let norm = w.iter().map(|v| v * v).sum::().sqrt(); + w.map(|v| v / norm) + } + }; + let positions = self.positions(&f); + let allowed = positions + .iter() + .enumerate() + .all(|(index, &x)| self.allowed(index, x)); + allowed.then(|| self.at(&positions)) + }) + .collect(); + // the corners first, so that a point that rounding sets apart from one is left out + front.splice(0..0, corners.iter().copied()); + let front = distinct(front); + if front.len() >= points { + return front; + } + divisions += 1; + } + } + + // exactly `points` points for 2 objectives, spread by length along a dense sample of the + // feasible x₁, the ends of its intervals included + fn two_objectives(&self, points: usize) -> Vec<[f64; M]> { + let mut values: Vec = (0..=SAMPLES) + .map(|i| i as f64 / SAMPLES as f64) + .filter(|&x| self.allowed(0, x)) + .collect(); + for (low, high) in self.intervals() { + values.extend([low, high]); + } + let mut dense: Vec<[f64; 2]> = values + .into_iter() + .map(|x| { + let f = self.at(&[x]); + [f[0], f[1]] + }) + .collect(); + dense.sort_by(|p, q| p[0].total_cmp(&q[0]).then(q[1].total_cmp(&p[1]))); + dense.dedup(); + spread_by_length(&dense, points) + .into_iter() + .map(|p| std::array::from_fn(|j| p[j])) + .collect() + } +} + +// the points without those within 1e-12 in every objective of an earlier one, in their order. +// Points of the plane or the sphere dominate each other only when rounding sets apart two that +// are the same, and then by less than this: the rest are mutually non-dominated +fn distinct(points: Vec<[f64; M]>) -> Vec<[f64; M]> { + let mut order: Vec = (0..points.len()).collect(); + order.sort_by(|&i, &k| points[i][0].total_cmp(&points[k][0])); + let mut removed = vec![false; points.len()]; + for (place, &i) in order.iter().enumerate() { + for &k in &order[place + 1..] { + if points[k][0] - points[i][0] > 1e-12 { + break; + } + if (0..M).all(|j| (points[k][j] - points[i][j]).abs() <= 1e-12) { + removed[i.max(k)] = true; + } + } + } + let kept = points.into_iter().zip(removed); + kept.filter_map(|(point, removed)| (!removed).then_some(point)) + .collect() +} + +macro_rules! dc_dtlz { + ( + $(#[$doc:meta])* + $name:ident, $label:literal, $shape:expr, $kind:expr, $k:literal, $b:literal + ) => { + $(#[$doc])* + #[derive(Clone, Copy, Debug, PartialEq)] + pub struct $name { + core: Core, + } + + impl $name { + /// The parameter a of genoxide's default problem. + pub const A: f64 = 3.0; + + /// The parameter b of genoxide's default problem. + pub const B: f64 = $b; + + #[doc = concat!("The problem with `variables` variables, at least `M`, a = 3 and b = ", stringify!($b), ".")] + /// + /// # Panics + /// + /// With fewer than 2 objectives, or fewer variables than objectives. + pub fn new(variables: usize) -> Self { + Self::with_parameters(variables, Self::A, Self::B) + } + + /// The problem with `variables` variables, at least `M`, and the parameters `a` + /// (the number of feasible segments or local optima, above 0) and `b` (how narrow + /// they are, between −1 and 1). + /// + /// # Panics + /// + /// With fewer than 2 objectives, fewer variables than objectives, an `a` that isn't + /// finite and above 0, or a `b` that isn't between −1 and 1. + pub fn with_parameters(variables: usize, a: f64, b: f64) -> Self { + Self { + core: Core::new($label, $shape, $kind, variables, a, b), + } + } + + /// The number of variables. + pub fn variables(&self) -> usize { + self.core.variables + } + + /// The parameter a. + pub fn a(&self) -> f64 { + self.core.a + } + + /// The parameter b. + pub fn b(&self) -> f64 { + self.core.b + } + } + + impl Default for $name { + #[doc = concat!("The problem with `M + ", stringify!($k), " − 1` variables, a = 3 and b = ", stringify!($b), ".")] + /// + /// # Panics + /// + /// With fewer than 2 objectives. + fn default() -> Self { + Self::new(M + $k - 1) + } + } + + impl MultiFitnessFunction for $name { + type Output = ([f64; M], f64); + + /// The objective values of `x` and its constraint violation, 0 when it's feasible. + /// + /// # Panics + /// + /// If `x` has fewer than `M` genes. + fn evaluate(&self, x: &Reals) -> ([f64; M], f64) { + self.core.evaluate(x) + } + } + + impl MultiProblem for $name { + type Representation = Real; + + fn name(&self) -> &'static str { + $label + } + + fn representation(&self) -> Real { + Real::uniform(self.core.variables, 0.0..=1.0).expect("valid bounds") + } + + fn reference(&self) -> &'static str { + REFERENCE + } + + fn reference_url(&self) -> Option<&'static str> { + Some(REFERENCE_URL) + } + + fn constraint_count(&self) -> usize { + self.core.count() + } + + fn constraints(&self, genome: &Reals) -> Constraints { + Constraints::new(self.core.values(genome), Vec::new()) + } + + fn optimal_front(&self, points: usize) -> Option> { + Some(self.core.front(points)) + } + + fn ideal_point(&self) -> Option<[f64; M]> { + Some(self.core.ideal()) + } + + fn nadir_point(&self) -> Option<[f64; M]> { + Some(self.core.nadir()) + } + } + }; +} + +dc_dtlz!( + /// DC1-DTLZ1: [`Dtlz1`] subject to `cos(aπx₁) ≥ b`, with a = 3 and b = 0.5 (the supplement's + /// eq. 12): feasible cones from the origin. + /// + /// With `M` objectives and `n` variables in [0, 1], `M + 4` by default. With a = 3 and + /// b = 0.5, x₁ is feasible in [0, 1/9] and [5/9, 7/9], a third of its range, and the front is + /// the parts of DTLZ1's (the plane `Σ fᵢ = 1/2`) with `f_M = (1 − x₁)/2` in [1/9, 2/9] and + /// [4/9, 1/2]: two bands, from the corner (0, …, 0, 1/2), each a cone from the origin through + /// the plane. The ideal point is (0, …, 0, 1/9) and the nadir point (7/18, …, 7/18, 1/2). + /// DTLZ1's g has 11ᵏ − 1 local fronts, all feasible where x₁ is. + /// + /// [`constraints`](MultiProblem::constraints) gives `b − cos(aπx₁)`. + /// + /// Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for + /// constrained multiobjective optimization. *IEEE Transactions on Evolutionary Computation* + /// 23(2): 303-315, supplementary document, section 1.2.1, eq. 12; see the module docs for + /// what was checked where, and what genoxide settles that the paper doesn't. + Dc1Dtlz1, + "DC1-DTLZ1", + Shape::Plane, + Kind::One, + 5, + 0.5 +); + +dc_dtlz!( + /// DC1-DTLZ3: [`Dtlz3`] subject to `cos(aπx₁) ≥ b`, with a = 3 and b = 0.5 (the supplement's + /// eq. 12): feasible cones from the origin. + /// + /// With `M` objectives and `n` variables in [0, 1], `M + 9` by default. With a = 3 and + /// b = 0.5, x₁ is feasible in [0, 1/9] and [5/9, 7/9], and the front is the parts of DTLZ3's + /// (the unit sphere) with `f_M = sin(πx₁/2)` in [0, sin(π/18)] and [sin(5π/18), sin(7π/18)]: + /// two bands, one along the base of the sphere. The ideal point is the origin and the nadir + /// point (1, …, 1, sin(7π/18)). DTLZ3's g has 3ᵏ − 1 local fronts (Deb et al.), spheres of + /// larger radii, all feasible where x₁ is. + /// + /// [`constraints`](MultiProblem::constraints) gives `b − cos(aπx₁)`. + /// + /// Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for + /// constrained multiobjective optimization. *IEEE Transactions on Evolutionary Computation* + /// 23(2): 303-315, supplementary document, section 1.2.1, eq. 12 (see [`Dc1Dtlz1`]). + Dc1Dtlz3, + "DC1-DTLZ3", + Shape::Sphere, + Kind::One, + 10, + 0.5 +); + +dc_dtlz!( + /// DC2-DTLZ1: [`Dtlz1`] subject to `cos(aπg) ≥ b` and `e^(−g) ≥ b`, with a = 3 and b = 0.9 + /// (the supplement's eqs. 13 and 14; b from the authors' lab's code): DTLZ1's front, and + /// almost nothing else feasible. + /// + /// With `M` objectives and `n` variables in [0, 1], `M + 4` by default. With b = 0.9, the + /// second constraint asks g ≤ −ln 0.9 ≈ 0.105 and the first `3πg ≤ arccos 0.9`, g ≤ 0.0479, + /// in the first of its bands: the feasible solutions are those within g ≈ 0.048 of the front, + /// where DTLZ1's g reaches 125 k. The front is DTLZ1's, whole: `Σ fᵢ = 1/2`, ideal point the + /// origin and nadir point (1/2, …, 1/2). The violation rises and falls with g (the + /// supplement's figure 6): its local minima, where `cos(aπg)` returns to 1, at g = 2/3, + /// 4/3, …, hold a population back. + /// + /// [`constraints`](MultiProblem::constraints) gives `b − cos(aπg)` and `b − e^(−g)`. + /// + /// Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for + /// constrained multiobjective optimization. *IEEE Transactions on Evolutionary Computation* + /// 23(2): 303-315, supplementary document, section 1.2.2, eqs. 13 and 14 (see [`Dc1Dtlz1`]). + Dc2Dtlz1, + "DC2-DTLZ1", + Shape::Plane, + Kind::Two, + 5, + 0.9 +); + +dc_dtlz!( + /// DC2-DTLZ3: [`Dtlz3`] subject to `cos(aπg) ≥ b` and `e^(−g) ≥ b`, with a = 3 and b = 0.5 + /// (the supplement's eqs. 13 and 14; b from the authors' lab's code): DTLZ3's front, and + /// almost nothing else feasible. + /// + /// With `M` objectives and `n` variables in [0, 1], `M + 9` by default. With b = 0.5, g must + /// be in [0, 1/9] or [5/9, ln 2]: spheres of radius 1 to 10/9 and 14/9 to 1.69 are + /// feasible. The front is DTLZ3's, whole: the unit sphere, ideal point the origin and nadir + /// point (1, …, 1). The violation's local minima, where `cos(aπg)` returns to 1, at + /// g = 2/3, 4/3, …, hold a population back. + /// + /// [`constraints`](MultiProblem::constraints) gives `b − cos(aπg)` and `b − e^(−g)`. + /// + /// Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for + /// constrained multiobjective optimization. *IEEE Transactions on Evolutionary Computation* + /// 23(2): 303-315, supplementary document, section 1.2.2, eqs. 13 and 14 (see [`Dc1Dtlz1`]). + Dc2Dtlz3, + "DC2-DTLZ3", + Shape::Sphere, + Kind::Two, + 10, + 0.5 +); + +dc_dtlz!( + /// DC3-DTLZ1: [`Dtlz1`] subject to `cos(aπxⱼ) ≥ b` for each position variable, + /// j = 1, …, M − 1, and `cos(aπg) ≥ b`, with a = 3 and b = 0.5 (the supplement's eqs. 15 and + /// 16, read as the module docs say): the front in patches, behind DC2's local optima. + /// + /// With `M` objectives and `n` variables in [0, 1], `M + 4` by default: M constraints. With + /// a = 3 and b = 0.5, every position variable is feasible in [0, 1/9] and [5/9, 7/9], and g in + /// [0, 1/9], [5/9, 7/9], [11/9, 13/9], …. The front is the parts of DTLZ1's where all the + /// position variables are: 2^(M−1) patches, some of them thin. The ideal point is + /// (0, …, 0, 1/9) and the nadir point (7/18 (7/9)^(M−2), …, 7/18, 1/2), the first objective + /// being the product of all the position variables. + /// + /// [`constraints`](MultiProblem::constraints) gives `b − cos(aπxⱼ)` for j < M, then + /// `b − cos(aπg)`. + /// + /// Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for + /// constrained multiobjective optimization. *IEEE Transactions on Evolutionary Computation* + /// 23(2): 303-315, supplementary document, section 1.2.3, eqs. 15 and 16 (see [`Dc1Dtlz1`]). + Dc3Dtlz1, + "DC3-DTLZ1", + Shape::Plane, + Kind::Three, + 5, + 0.5 +); + +dc_dtlz!( + /// DC3-DTLZ3: [`Dtlz3`] subject to `cos(aπxⱼ) ≥ b` for each position variable, + /// j = 1, …, M − 1, and `cos(aπg) ≥ b`, with a = 3 and b = 0.5 (the supplement's eqs. 15 and + /// 16, read as the module docs say): the front in patches, behind DC2's local optima. + /// + /// With `M` objectives and `n` variables in [0, 1], `M + 9` by default: M constraints. With + /// a = 3 and b = 0.5, every position variable is feasible in [0, 1/9] and [5/9, 7/9], and the + /// front is the parts of DTLZ3's, the unit sphere, where all of them are: 2^(M−1) patches. + /// The ideal point is (cos(7π/18)^(M−1), 0, …, 0) and the nadir point + /// (1, sin(7π/18), …, sin(7π/18)): f₁, a product of cosines, is 1 with every position variable + /// at 0 and least with all of them at 7/9, and each other objective has a sine factor, of a + /// variable at most 7/9. + /// + /// [`constraints`](MultiProblem::constraints) gives `b − cos(aπxⱼ)` for j < M, then + /// `b − cos(aπg)`. + /// + /// Li, K., Chen, R., Fu, G. and Yao, X. (2019). Two-archive evolutionary algorithm for + /// constrained multiobjective optimization. *IEEE Transactions on Evolutionary Computation* + /// 23(2): 303-315, supplementary document, section 1.2.3, eqs. 15 and 16 (see [`Dc1Dtlz1`]). + Dc3Dtlz3, + "DC3-DTLZ3", + Shape::Sphere, + Kind::Three, + 10, + 0.5 +); + +/// The DC-DTLZ problems with `M` objectives at their default sizes, for [`all`](super::all). +pub(super) fn all() -> Vec>> { + vec![ + boxed(Dc1Dtlz1::::default()), + boxed(Dc1Dtlz3::::default()), + boxed(Dc2Dtlz1::::default()), + boxed(Dc2Dtlz3::::default()), + boxed(Dc3Dtlz1::::default()), + boxed(Dc3Dtlz3::::default()), + ] +} + +#[cfg(test)] +mod tests { + use super::*; + use crate::Objective::Minimize; + use crate::StreamRng; + use crate::genome::Representation; + use crate::multi::{Scores, non_dominated_sort}; + + fn at(values: &[f64]) -> Reals { + Reals::from(values.to_vec()) + } + + fn assert_close(actual: &[f64], expected: &[f64]) { + assert_eq!(actual.len(), expected.len()); + for (a, e) in actual.iter().zip(expected) { + assert!( + (a - e).abs() <= 1e-12 * e.abs().max(1.0), + "{actual:?} is not {expected:?}" + ); + } + } + + // the formulas of the supplement, at points chosen so that the values follow by hand + #[test] + fn values_match_the_supplement() { + // DC1-DTLZ1 (M = 3, n = 7): x₁ = 0 is feasible, cos 0 = 1 ≥ 0.5; x₁ = 1/3 isn't, + // cos π = −1, by 1.5; the objectives are DTLZ1's + let problem = Dc1Dtlz1::<3>::default(); + assert_eq!(problem.variables(), 7); + let x = at(&[0.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5]); + assert_eq!(problem.evaluate(&x), ([0.0, 0.0, 0.5], 0.0)); + assert_close(problem.constraints(&x).inequalities(), &[-0.5]); + let x = at(&[1.0 / 3.0, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5]); + let (f, violation) = problem.evaluate(&x); + assert_close(&f, &[1.0 / 12.0, 1.0 / 12.0, 1.0 / 3.0]); + assert_close(&[violation], &[1.5]); + + // DC2-DTLZ1 (b = 0.9): g = 0 at the distance variables at 0.5, feasible by 0.1 twice; a + // distance variable at 0.6 adds 100 (0.01 − cos 2π + 1) = 1 to g, cos 3π = −1 and + // e^(−1): violations 1.9 and 0.9 − e^(−1) + let problem = Dc2Dtlz1::<3>::default(); + assert_eq!((problem.a(), problem.b()), (3.0, 0.9)); + let x = at(&[0.5; 7]); + assert_close(problem.constraints(&x).inequalities(), &[-0.1, -0.1]); + let x = at(&[0.5, 0.5, 0.6, 0.5, 0.5, 0.5, 0.5]); + let (f, violation) = problem.evaluate(&x); + assert_close(&f, &[0.25, 0.25, 0.5]); + assert_close(&[violation], &[1.9 + 0.9 - math::exp(-1.0)]); + + // DC2-DTLZ3 (b = 0.5, n = 12): g = 2/3 at cos(2π) = 1 is a local minimum of the first + // violation, where e^(−2/3) ≈ 0.513 ≥ 0.5: feasible. Two distance variables at + // 0.5 ± d with 100 (d² − cos 20πd + 1) = 1/3 each give it + let problem = Dc2Dtlz3::<3>::default(); + assert_eq!(problem.variables(), 12); + // d solves d² − cos(20πd) + 1 = 1/300, by bisection + let h = |d: f64| d * d - math::cos(20.0 * PI * d) + 1.0 - 1.0 / 300.0; + let (mut low, mut high) = (0.0, 0.01); + for _ in 0..100 { + let middle = 0.5 * (low + high); + if h(middle) < 0.0 { + low = middle; + } else { + high = middle; + } + } + let mut genes = vec![0.5; 12]; + genes[2] = 0.5 + low; + genes[3] = 0.5 - low; + let x = at(&genes); + let g = rastrigin_g(&genes[2..]); + assert!((g - 2.0 / 3.0).abs() < 1e-9, "{g}"); + let values = problem.constraints(&x).inequalities().to_vec(); + assert!((values[0] + 0.5).abs() < 1e-6, "{values:?}"); + assert_close(&values[1..], &[0.5 - math::exp(-g)]); + assert_eq!(problem.evaluate(&x).1, 0.0); + + // DC3-DTLZ1 (M = 3): constraints on x₁, x₂ and g; (0.6, 0.7) is feasible in both, + // (0.6, 0.3) not in x₂: 0.5 − cos(0.9π) + let problem = Dc3Dtlz1::<3>::default(); + assert_eq!(problem.constraint_count(), 3); + let x = at(&[0.6, 0.7, 0.5, 0.5, 0.5, 0.5, 0.5]); + assert_eq!(problem.evaluate(&x).1, 0.0); + let x = at(&[0.6, 0.3, 0.5, 0.5, 0.5, 0.5, 0.5]); + let values = problem.constraints(&x).inequalities().to_vec(); + assert_close(&values[1..], &[0.5 - math::cos(0.9 * PI), -0.5]); + assert_close(&[problem.evaluate(&x).1], &[0.5 - math::cos(0.9 * PI)]); + + // DC3-DTLZ3 with other parameters: a = 2, b = 0 + let problem = Dc3Dtlz3::<3>::with_parameters(12, 2.0, 0.0); + assert_eq!((problem.a(), problem.b()), (2.0, 0.0)); + let x = at(&genes); + assert_close( + &problem.constraints(&x).inequalities()[..2], + &[-math::cos(PI), -math::cos(PI)], + ); + } + + #[test] + fn feasible_values_of_a_variable() { + let core = Dc1Dtlz1::<3>::default().core; + let intervals = core.intervals(); + assert_eq!(intervals.len(), 2); + let expected = [(0.0, 1.0 / 9.0), (5.0 / 9.0, 7.0 / 9.0)]; + for ((low, high), (a, b)) in intervals.iter().zip(expected) { + assert!((low - a).abs() < 1e-12 && (high - b).abs() < 1e-12); + assert!(core.cosine(*low) <= 0.0 && core.cosine(*high) <= 0.0); + } + // the ends: the last feasible floating-point values + let (_, high) = intervals[1]; + assert!(core.cosine(next_up(high)) > 0.0); + // with a = 2 and b = 0, x = 1 is feasible: [0, 1/4], [3/4, 1] + let core = Dc1Dtlz1::<3>::with_parameters(7, 2.0, 0.0).core; + let intervals = core.intervals(); + assert_eq!(intervals.len(), 2); + assert_eq!(intervals[1].1, 1.0); + assert!((intervals[0].1 - 0.25).abs() < 1e-12); + } + + fn next_up(x: f64) -> f64 { + f64::from_bits(x.to_bits() + 1) + } + + // the fronts: on DTLZ's, with feasible genomes, mutually non-dominated, between the ideal + // and nadir points that they reach + fn check_front( + problem: &P, + points: usize, + on_front: impl Fn(&[f64; M]) -> bool, + ) where + P: MultiProblem + + MultiFitnessFunction, + { + let front = problem.optimal_front(points).expect("known"); + if M == 2 { + assert_eq!(front.len(), points); + } else { + assert!(front.len() >= points); + } + let scores: Vec> = front.iter().map(|f| Scores::new(*f)).collect(); + assert_eq!(non_dominated_sort(&scores, &[Minimize; M]).len(), 1); + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + for f in &front { + assert!(on_front(f), "{}: {f:?}", problem.name()); + for j in 0..M { + assert!(f[j] >= ideal[j] && f[j] <= nadir[j], "{f:?}"); + } + } + for j in 0..M { + assert!(front.iter().any(|f| f[j] == ideal[j]), "{}", problem.name()); + assert!(front.iter().any(|f| f[j] == nadir[j]), "{}", problem.name()); + } + } + + #[test] + fn the_fronts_lie_where_derived() { + let plane = |f: &[f64; 3]| (f.iter().sum::() - 0.5).abs() < 1e-12; + let sphere = |f: &[f64; 3]| (f.iter().map(|v| v * v).sum::() - 1.0).abs() < 1e-12; + // DC1: f₃ = (1 − x₁)/2 in [1/9, 2/9] or [4/9, 1/2] (DTLZ1), sin(πx₁/2) in [0, sin(π/18)] + // or [sin(5π/18), sin(7π/18)] (DTLZ3) + let near = |v: f64, low: f64, high: f64| v >= low - 1e-12 && v <= high + 1e-12; + check_front(&Dc1Dtlz1::<3>::default(), 200, |f| { + plane(f) && (near(f[2], 1.0 / 9.0, 2.0 / 9.0) || near(f[2], 4.0 / 9.0, 0.5)) + }); + let s = |v: f64| math::sin(PI * v / 18.0); + check_front(&Dc1Dtlz3::<3>::default(), 200, |f| { + sphere(f) && (near(f[2], 0.0, s(1.0)) || near(f[2], s(5.0), s(7.0))) + }); + check_front(&Dc2Dtlz1::<3>::default(), 91, plane); + check_front(&Dc2Dtlz3::<3>::default(), 91, sphere); + let dc3 = Dc3Dtlz1::<3>::default(); + check_front(&dc3, 200, |f| { + plane(f) + && dc3 + .core + .positions(f) + .iter() + .all(|&x| near(x, 0.0, 1.0 / 9.0) || near(x, 5.0 / 9.0, 7.0 / 9.0)) + }); + check_front(&Dc3Dtlz3::<3>::default(), 200, sphere); + // the ideal and nadir points of the docs + assert_close( + &Dc1Dtlz1::<3>::default().ideal_point().unwrap(), + &[0.0, 0.0, 1.0 / 9.0], + ); + assert_close( + &Dc1Dtlz1::<3>::default().nadir_point().unwrap(), + &[7.0 / 18.0, 7.0 / 18.0, 0.5], + ); + assert_close( + &Dc1Dtlz3::<3>::default().nadir_point().unwrap(), + &[1.0, 1.0, s(7.0)], + ); + assert_close( + &Dc3Dtlz1::<3>::default().nadir_point().unwrap(), + &[7.0 / 18.0 * 7.0 / 9.0, 7.0 / 18.0, 0.5], + ); + // two objectives: exactly the points asked for, with both ends of each piece + let problem = Dc1Dtlz1::<2>::default(); + check_front(&problem, 50, |f| (f[0] + f[1] - 0.5).abs() < 1e-12); + check_front(&Dc3Dtlz3::<2>::default(), 37, |f| { + (f[0] * f[0] + f[1] * f[1] - 1.0).abs() < 1e-12 + }); + let front = problem.optimal_front(1_000).unwrap(); + let x1 = |f: &[f64; 2]| 2.0 * f[0]; + assert!( + front + .iter() + .all(|f| near(x1(f), 0.0, 1.0 / 9.0) || near(x1(f), 5.0 / 9.0, 7.0 / 9.0)) + ); + // more objectives + check_front(&Dc1Dtlz1::<4>::default(), 100, |f| { + (f.iter().sum::() - 0.5).abs() < 1e-12 + }); + check_front(&Dc3Dtlz3::<4>::default(), 100, |f| { + (f.iter().map(|v| v * v).sum::() - 1.0).abs() < 1e-12 + }); + } + + // no feasible solution is better than the front: random genomes with their distance + // variables near 0.5, every feasible one weakly dominated by a point of the front, up to its + // spacing, and dominating none + fn check_against_random

(problem: &P, noise: f64, tolerance: f64) + where + P: MultiProblem<3, Representation = Real> + + MultiFitnessFunction, + { + let front = problem.optimal_front(5_000).expect("known"); + let mut rng = StreamRng::seed_from_u64(5); + let real = problem.representation(); + let mut feasible = 0; + for _ in 0..10_000 { + let mut genome = real.random_genome(&mut rng).to_vec(); + for gene in &mut genome[2..] { + *gene = 0.5 + (*gene - 0.5) * 2.0 * noise; + } + let (f, violation) = problem.evaluate(&at(&genome)); + if violation > 0.0 { + continue; + } + feasible += 1; + assert!( + front + .iter() + .any(|q| (0..3).all(|j| q[j] <= f[j] + tolerance)), + "{}: {f:?}", + problem.name() + ); + for q in &front { + assert!( + !(0..3).all(|j| f[j] < q[j] - 1e-9), + "{}: {f:?} dominates {q:?}", + problem.name() + ); + } + } + assert!(feasible > 50, "{}: {feasible}", problem.name()); + } + + #[test] + fn random_solutions_agree_with_the_fronts() { + check_against_random(&Dc1Dtlz1::<3>::default(), 0.001, 0.02); + check_against_random(&Dc1Dtlz3::<3>::default(), 0.001, 0.03); + check_against_random(&Dc2Dtlz1::<3>::default(), 0.0005, 0.02); + check_against_random(&Dc2Dtlz3::<3>::default(), 0.001, 0.03); + check_against_random(&Dc3Dtlz1::<3>::default(), 0.001, 0.02); + check_against_random(&Dc3Dtlz3::<3>::default(), 0.001, 0.03); + } + + #[test] + fn sizes_and_parameters() { + assert_eq!(Dc1Dtlz3::<3>::default().variables(), 12); + assert_eq!(Dc2Dtlz1::<5>::default().variables(), 9); + assert_eq!(Dc3Dtlz3::<5>::default().constraint_count(), 5); + assert_eq!(Dc2Dtlz3::<3>::default().constraint_count(), 2); + assert_eq!(Dc1Dtlz1::<3>::B, 0.5); + assert_eq!(Dc2Dtlz1::<3>::B, 0.9); + let names: Vec<_> = all::<3>().iter().map(|p| p.name()).collect(); + assert_eq!( + names, + [ + "DC1-DTLZ1", + "DC1-DTLZ3", + "DC2-DTLZ1", + "DC2-DTLZ3", + "DC3-DTLZ1", + "DC3-DTLZ3" + ] + ); + } + + #[test] + #[should_panic(expected = "a finite b between -1 and 1")] + fn a_b_of_1_panics() { + let _ = Dc1Dtlz1::<3>::with_parameters(7, 3.0, 1.0); + } +} diff --git a/src/multi/problems/dtlz_constrained.rs b/src/multi/problems/dtlz_constrained.rs new file mode 100644 index 00000000..ce0c46e3 --- /dev/null +++ b/src/multi/problems/dtlz_constrained.rs @@ -0,0 +1,575 @@ +//! DTLZ8 and DTLZ9: the constrained problems of Deb, Thiele, Laumanns and Zitzler's report, built +//! by its constraint surface approach: objectives that each average or sum their own block of +//! variables, and constraints on the objectives that cut the front out of their space. +//! +//! Deb, K., Thiele, L., Laumanns, M. and Zitzler, E. (2001). *Scalable Test Problems for +//! Evolutionary Multi-Objective Optimization.* TIK-Report 112, Computer Engineering and Networks +//! Laboratory, ETH Zürich, sections 8.8 and 8.9, eqs. 28 and 29, read in the report (the 2005 book +//! chapter, *Evolutionary Multiobjective Optimization*, Springer: 105-145, which reprints it, +//! wasn't compared). The report's DTLZ8 is the 2002 conference paper's DTLZ7 and its DTLZ9 isn't +//! in the paper; genoxide follows the report's numbering, as for DTLZ5-7. +//! +//! **The blocks.** Each objective fⱼ takes the j-th of M blocks of variables, the report printing +//! its sum as running from `⌊(j − 1) n/M⌋` to `⌊j n/M⌋`. With 1-based variables, as the report +//! counts them elsewhere, that would start the first block at a variable x₀ that doesn't exist +//! and let neighbouring blocks share a variable: the sums here run from `⌊(j − 1) n/M⌋ + 1` to +//! `⌊j n/M⌋`, which splits the n variables into M blocks of ⌊n/M⌋ or ⌈n/M⌉ each, all of them +//! ⌊n/M⌋ = n/M for the report's n = 10M. No code from the report's authors was found to check +//! this reading against. +//! +//! The report gives no constraint-free fronts: the fronts here are derived from the constraints, +//! as each problem's docs show, and the tests check them against random feasible solutions. + +use super::{MultiProblem, evenly}; +use crate::constraint::at_most; +use crate::genome::{Real, Reals}; +use crate::math; +use crate::multi::MultiFitnessFunction; +use crate::problems::Constraints; +use std::f64::consts::FRAC_PI_2; + +const REPORT: &str = "Deb, K., Thiele, L., Laumanns, M. and Zitzler, E. (2001). Scalable Test \ + Problems for Evolutionary Multi-Objective Optimization. TIK-Report 112, \ + Computer Engineering and Networks Laboratory, ETH Zürich."; +const REPORT_URL: &str = "https://sop.tik.ee.ethz.ch/publicationListFiles/dtlz2001a.pdf"; + +// the total violation of constraints g(x) <= 0 +fn violation(values: &[f64]) -> f64 { + values.iter().map(|&g| at_most(g, 0.0)).sum() +} + +// the 0-based range of the variables of the j-th block (0-based j) of n variables in M blocks: +// the report's ⌊(j − 1) n/M⌋ + 1 to ⌊j n/M⌋, 1-based +fn block(j: usize, n: usize, m: usize) -> std::ops::Range { + j * n / m..(j + 1) * n / m +} + +/// DTLZ8: M objectives, each the mean of its own block of variables, subject to M constraints +/// that leave a front made of a line and a surface. +/// +/// ```text +/// minimize fⱼ = (1/⌊n/M⌋) Σ_{i in the j-th block} xᵢ, j = 1, …, M +/// subject to f_M + 4 fⱼ − 1 ≥ 0, j = 1, …, M − 1 +/// 2 f_M + min_{i≠j} (fᵢ + fⱼ) − 1 ≥ 0, i, j = 1, …, M − 1 +/// ``` +/// +/// With `M` objectives, from 3 on (the last constraint takes the least sum of two different +/// objectives other than `f_M`), and `n` variables in [0, 1], `10 M` by default, as the report +/// suggests; the j-th block is `x_{⌊(j−1)n/M⌋+1}` to `x_{⌊jn/M⌋}`. The report prints the sum +/// from `⌊(j − 1) n/M⌋`, which with 1-based variables would start the first block at a variable +/// x₀ that doesn't exist and let neighbouring blocks share one: genoxide reads it as above, M +/// blocks of ⌊n/M⌋ or ⌈n/M⌉ variables, n/M each for the report's n = 10 M. The objectives can take any values in [0, 1] independently, so the problem +/// is its constraints: the report's front is "a combination of a straight line and a +/// hyper-plane", the line where the first M − 1 constraints meet with f₁ = … = f_{M−1}, and the +/// last constraint's plane. +/// +/// **The front**, derived here: the line `f₁ = … = f_{M−1} = t`, `f_M = 1 − 4t` for t in +/// [0, 1/6], from (0, …, 0, 1) to (1/6, …, 1/6, 1/3), and from its end the part of the plane +/// `2 f_M + fᵢ + fⱼ = 1` where one objective s (any of the first M − 1) is at most all the +/// others, which are equal, u: `f_M = t` in [0, 1/3], s from `(1 − t)/4` (the first constraints) +/// to `(1 − 2t)/2` (s = u), and `u = 1 − 2t − s`. A point with two of the first M − 1 +/// objectives above the least that differ is dominated by the one with both at the lesser, +/// feasible still; for M = 3 the plane part is the triangle from (1/6, 1/6, 1/3) to (1/4, 3/4, 0) +/// and (3/4, 1/4, 0). The ideal point is the origin, and the nadir point (3/4, …, 3/4, 1). +/// +/// [`constraints`](MultiProblem::constraints) gives `1 − f_M − 4 fⱼ` for j < M, then +/// `1 − 2 f_M − min_{i≠j} (fᵢ + fⱼ)`. +/// +/// The report's NSGA-II and SPEA2 runs (figures 28 and 29) find parts of the front with many +/// solutions on the surfaces next to it, which the front only weakly dominates. +/// +/// Deb, K., Thiele, L., Laumanns, M. and Zitzler, E. (2001). *Scalable Test Problems for +/// Evolutionary Multi-Objective Optimization.* TIK-Report 112, ETH Zürich, section 8.8, eq. 28. +#[derive(Clone, Copy, Debug, PartialEq, Eq, Hash)] +pub struct Dtlz8 { + variables: usize, +} + +impl Dtlz8 { + /// The number of constraints: one per objective. + pub const CONSTRAINTS: usize = M; + + /// The problem with `variables` variables, at least `M`. + /// + /// # Panics + /// + /// With fewer than 3 objectives, or fewer variables than objectives. + pub fn new(variables: usize) -> Self { + assert!(M >= 3, "DTLZ8 needs at least 3 objectives"); + assert!( + variables >= M, + "DTLZ8 needs at least as many variables as objectives" + ); + Self { variables } + } + + /// The number of variables. + pub fn variables(&self) -> usize { + self.variables + } + + // the objectives: the mean of each block, divided by ⌊n/M⌋ as the report prints it + fn objectives(&self, x: &Reals) -> [f64; M] { + let n = x.len(); + let size = (n / M) as f64; + std::array::from_fn(|j| x[block(j, n, M)].iter().sum::() / size) + } + + // the constraints as g <= 0, from the objectives + fn values(f: &[f64; M]) -> [f64; M] { + let last = f[M - 1]; + // the least sum of two different objectives other than f_M: the two least of them + let (mut least, mut second) = (f64::INFINITY, f64::INFINITY); + for &fi in &f[..M - 1] { + if fi < least { + (least, second) = (fi, least); + } else if fi < second { + second = fi; + } + } + std::array::from_fn(|j| { + if j < M - 1 { + 1.0 - last - 4.0 * f[j] + } else { + 1.0 - 2.0 * last - (least + second) + } + }) + } + + // the front with the resolution k: k points of the line, and the plane part on a triangular + // grid of k + 1 rows in t, for each objective that can be the least + fn grid(k: usize) -> Vec<[f64; M]> { + let mut front = Vec::new(); + for i in 0..k { + let t = i as f64 / k as f64 / 6.0; + let mut f = [t; M]; + f[M - 1] = 1.0 - 4.0 * t; + front.push(f); + } + for least in 0..M - 1 { + for i in 0..=k { + let t = i as f64 / k as f64 / 3.0; + let low = (1.0 - t) / 4.0; + let width = (1.0 - 3.0 * t) / 4.0; + let steps = k - i; + for j in 0..=steps { + // s = u (the end of a row) is the same point for every choice of the least + if least > 0 && j == steps { + continue; + } + let s = if steps == 0 { + low + } else { + low + width * j as f64 / steps as f64 + }; + let u = 1.0 - 2.0 * t - s; + let mut f = [u; M]; + f[least] = s; + f[M - 1] = t; + front.push(f); + } + } + } + front + } + + fn front(points: usize) -> Vec<[f64; M]> { + if points == 0 { + return Vec::new(); + } + let mut k = 1; + loop { + let front = Self::grid(k); + if front.len() >= points { + return front; + } + k += 1; + } + } +} + +/// DTLZ9: M objectives, each the sum of `xᵢ^0.1` over its own block of variables, subject to +/// M − 1 constraints that leave a front that is a curve. +/// +/// ```text +/// minimize fⱼ = Σ_{i in the j-th block} xᵢ^0.1, j = 1, …, M +/// subject to f_M² + fⱼ² − 1 ≥ 0, j = 1, …, M − 1 +/// ``` +/// +/// With `M` objectives, from 2 on, and `n` variables in [0, 1], `10 M` by default, as the report +/// suggests; the j-th block is `x_{⌊(j−1)n/M⌋+1}` to `x_{⌊jn/M⌋}`, read as for [`Dtlz8`]. +/// The report prints no mean here, unlike DTLZ8: an objective ranges over [0, ⌈n/M⌉]. +/// +/// **The front** is the curve `f₁ = … = f_{M−1} = cos θ`, `f_M = sin θ` for θ in [0, π/2], where +/// all the constraints meet, as the report says (derived here: for a given `f_M` in [0, 1], each +/// constraint asks `fⱼ ≥ √(1 − f_M²)` alone). With f_M and any other objective, it's the unit +/// circle's quarter; with two others, the diagonal. Its ideal point is the origin and its nadir +/// point (1, …, 1). The power 0.1 makes small objectives hard to reach: on the front each +/// variable is about `(fⱼ/10)^10`, below 10⁻¹⁰, and a random solution is near the top of its +/// range, the report's "the density of solutions gets thinner towards the Pareto-optimal +/// region". +/// +/// [`constraints`](MultiProblem::constraints) gives `1 − f_M² − fⱼ²` for j < M. +/// +/// Deb, K., Thiele, L., Laumanns, M. and Zitzler, E. (2001). *Scalable Test Problems for +/// Evolutionary Multi-Objective Optimization.* TIK-Report 112, ETH Zürich, section 8.9, eq. 29. +#[derive(Clone, Copy, Debug, PartialEq, Eq, Hash)] +pub struct Dtlz9 { + variables: usize, +} + +impl Dtlz9 { + /// The number of constraints: one per objective but the last. + pub const CONSTRAINTS: usize = M - 1; + + /// The problem with `variables` variables, at least `M`. + /// + /// # Panics + /// + /// With fewer than 2 objectives, or fewer variables than objectives. + pub fn new(variables: usize) -> Self { + assert!(M >= 2, "DTLZ9 needs at least 2 objectives"); + assert!( + variables >= M, + "DTLZ9 needs at least as many variables as objectives" + ); + Self { variables } + } + + /// The number of variables. + pub fn variables(&self) -> usize { + self.variables + } + + fn objectives(&self, x: &Reals) -> [f64; M] { + let n = x.len(); + std::array::from_fn(|j| { + x[block(j, n, M)] + .iter() + .map(|&xi| math::powf(xi, 0.1)) + .sum() + }) + } + + fn values(f: &[f64; M]) -> Vec { + let last = f[M - 1] * f[M - 1]; + f[..M - 1].iter().map(|fj| 1.0 - last - fj * fj).collect() + } + + // `points` points of the curve, evenly spread in angle + fn front(points: usize) -> Vec<[f64; M]> { + (0..points) + .map(|i| { + let (sin, cos) = math::sin_cos(FRAC_PI_2 * evenly(i, points)); + let mut f = [cos; M]; + f[M - 1] = sin; + f + }) + .collect() + } +} + +// Default, the fitness and the metadata +macro_rules! constraint_surface { + ($name:ident, $label:literal, $count:expr, $ideal:expr, $nadir:expr) => { + impl Default for $name { + /// The report's problem, with `10 M` variables. + /// + /// # Panics + /// + /// Where [`new`](Self::new) panics. + fn default() -> Self { + Self::new(10 * M) + } + } + + impl MultiFitnessFunction for $name { + type Output = ([f64; M], f64); + + /// The objective values of `x` and its constraint violation, 0 when it's feasible. + /// + /// # Panics + /// + /// If `x` has fewer than `M` genes. + fn evaluate(&self, x: &Reals) -> ([f64; M], f64) { + let f = self.objectives(x); + (f, violation(&Self::values(&f))) + } + } + + impl MultiProblem for $name { + type Representation = Real; + + fn name(&self) -> &'static str { + $label + } + + fn representation(&self) -> Real { + Real::uniform(self.variables, 0.0..=1.0).expect("valid bounds") + } + + fn reference(&self) -> &'static str { + REPORT + } + + fn reference_url(&self) -> Option<&'static str> { + Some(REPORT_URL) + } + + fn constraint_count(&self) -> usize { + $count + } + + fn constraints(&self, genome: &Reals) -> Constraints { + let f = self.objectives(genome); + Constraints::new(Self::values(&f).to_vec(), Vec::new()) + } + + fn optimal_front(&self, points: usize) -> Option> { + Some(Self::front(points)) + } + + fn ideal_point(&self) -> Option<[f64; M]> { + Some($ideal) + } + + fn nadir_point(&self) -> Option<[f64; M]> { + Some($nadir) + } + } + }; +} + +constraint_surface!(Dtlz8, "DTLZ8", M, [0.0; M], { + let mut nadir = [0.75; M]; + nadir[M - 1] = 1.0; + nadir +}); +constraint_surface!(Dtlz9, "DTLZ9", M - 1, [0.0; M], [1.0; M]); + +#[cfg(test)] +mod tests { + use super::*; + use crate::Objective::Minimize; + use crate::StreamRng; + use crate::genome::Representation; + use crate::multi::{Scores, non_dominated_sort}; + + fn at(values: &[f64]) -> Reals { + Reals::from(values.to_vec()) + } + + fn assert_close(actual: &[f64], expected: &[f64]) { + assert_eq!(actual.len(), expected.len()); + for (a, e) in actual.iter().zip(expected) { + assert!( + (a - e).abs() <= 1e-12 * e.abs().max(1.0), + "{actual:?} is not {expected:?}" + ); + } + } + + #[test] + fn blocks_split_the_variables() { + // the report's n = 10M: blocks of 10 + assert_eq!(block(0, 30, 3), 0..10); + assert_eq!(block(2, 30, 3), 20..30); + // 7 variables in 3 blocks: ⌊7/3⌋ = 2, ⌊14/3⌋ = 4, 7 + assert_eq!( + (0..3).map(|j| block(j, 7, 3)).collect::>(), + [0..2, 2..4, 4..7] + ); + } + + // the formulas of the report, at points chosen so that the values follow by hand + #[test] + fn values_match_the_report() { + // DTLZ8, M = 3, n = 30: blocks at 1/8, 1/4 and 5/8 give (1/8, 1/4, 5/8); the constraints + // 1 − 5/8 − 1/2, 1 − 5/8 − 1 and 1 − 5/4 − 3/8, all feasible + let problem = Dtlz8::<3>::default(); + assert_eq!(problem.variables(), 30); + let mut genes = vec![0.125; 30]; + genes[10..20].fill(0.25); + genes[20..].fill(0.625); + let x = at(&genes); + let (f, violation) = problem.evaluate(&x); + assert_close(&f, &[0.125, 0.25, 0.625]); + assert_eq!(violation, 0.0); + assert_close( + problem.constraints(&x).inequalities(), + &[-0.125, -0.625, -0.625], + ); + // a mean, not a sum: (1/16, 1/8, 1/4) breaks all three constraints, by 1 − 1/4 − 1/4, + // 1 − 1/4 − 1/2 and 1 − 1/2 − 3/16 + genes.iter_mut().for_each(|x| *x /= 2.0); + genes[20..].fill(0.25); + let (f, violation) = problem.evaluate(&at(&genes)); + assert_close(&f, &[0.0625, 0.125, 0.25]); + assert_close(&[violation], &[0.5 + 0.25 + 0.3125]); + // with 4 objectives, the least two of f₁…f₃ count: (0.3, 0.1, 0.2, 0.3) gives + // 1 − 0.6 − 0.3 = 0.1 + let f = [0.3, 0.1, 0.2, 0.3]; + assert_close(&Dtlz8::<4>::values(&f)[3..], &[1.0 - 0.6 - 0.3]); + + // DTLZ9, M = 3, n = 30: a block of 10 genes at 1 sums to 10; at 2⁻¹⁰ each gives 0.5 + let problem = Dtlz9::<3>::default(); + let mut genes = vec![1.0; 30]; + genes[..10].fill(0.5f64.powi(10)); + let x = at(&genes); + let (f, violation) = problem.evaluate(&x); + assert_close(&f, &[5.0, 10.0, 10.0]); + assert_eq!(violation, 0.0); + assert_close(problem.constraints(&x).inequalities(), &[-124.0, -199.0]); + // all at 0: the origin, breaking both constraints by 1 + let (f, violation) = problem.evaluate(&at(&[0.0; 30])); + assert_eq!(f, [0.0; 3]); + assert_eq!(violation, 2.0); + } + + // a genome of DTLZ8 or DTLZ9 whose blocks give the objectives f: DTLZ8 with each block at its + // mean, DTLZ9 with each variable at (fⱼ / its block's size)^10 + fn dtlz8_genome(f: &[f64; M], n: usize) -> Reals { + let mut genes = vec![0.0; n]; + for (j, &fj) in f.iter().enumerate() { + genes[block(j, n, M)].fill(fj); + } + Reals::from(genes) + } + + fn dtlz9_genome(f: &[f64; M], n: usize) -> Reals { + let mut genes = vec![0.0; n]; + for (j, &fj) in f.iter().enumerate() { + let range = block(j, n, M); + let size = range.len() as f64; + genes[range].fill(math::powi(fj / size, 10)); + } + Reals::from(genes) + } + + // the fronts: on the curves derived, feasible, reached by genomes, and mutually non-dominated + #[test] + fn the_fronts_lie_where_derived() { + fn dtlz8() { + let problem = Dtlz8::::default(); + let front = problem.optimal_front(300).expect("known"); + assert!(front.len() >= 300); + let scores: Vec> = front.iter().map(|f| Scores::new(*f)).collect(); + assert_eq!(non_dominated_sort(&scores, &[Minimize; M]).len(), 1); + for f in &front { + let values = Dtlz8::::values(f); + assert!(values.iter().all(|&g| g <= 1e-12), "{f:?}: {values:?}"); + // on the line where the first M − 1 constraints are tight, or on the last + // constraint's plane + let on_line = values[..M - 1].iter().all(|g| g.abs() < 1e-12); + assert!(on_line || values[M - 1].abs() < 1e-12, "{f:?}"); + let (genome_f, violation) = problem.evaluate(&dtlz8_genome(f, 10 * M)); + assert_close(&genome_f, f); + assert!(violation < 1e-12); + } + let ideal = problem.ideal_point().expect("known"); + let nadir = problem.nadir_point().expect("known"); + for j in 0..M { + let low = front.iter().map(|f| f[j]).fold(f64::INFINITY, f64::min); + let high = front.iter().map(|f| f[j]).fold(f64::NEG_INFINITY, f64::max); + assert_eq!((low, high), (ideal[j], nadir[j])); + } + } + dtlz8::<3>(); + dtlz8::<4>(); + dtlz8::<5>(); + // the line's end and the triangle's corners, for 3 objectives + let front = Dtlz8::<3>::grid(6); + for corner in [ + [1.0 / 6.0, 1.0 / 6.0, 1.0 / 3.0], + [0.25, 0.75, 0.0], + [0.75, 0.25, 0.0], + ] { + assert!( + front + .iter() + .any(|f| (0..3).all(|j| (f[j] - corner[j]).abs() < 1e-15)) + ); + } + // k = 1: the line's start, three points of the first row and the junction for f₁ the + // least, and one more for f₂ the least + assert_eq!(Dtlz8::<3>::grid(1).len(), 1 + 3 + 1); + + let problem = Dtlz9::<3>::default(); + let front = problem.optimal_front(100).expect("known"); + assert_eq!(front.len(), 100); + for f in &front { + assert!((f[0] * f[0] + f[2] * f[2] - 1.0).abs() < 1e-15 && f[0] == f[1]); + let (genome_f, violation) = problem.evaluate(&dtlz9_genome(f, 30)); + assert!((0..3).all(|j| (genome_f[j] - f[j]).abs() < 1e-12)); + assert!(violation < 1e-12); + } + assert_eq!(front[0], [1.0, 1.0, 0.0]); + assert_eq!(Dtlz9::<2>::default().optimal_front(7).unwrap().len(), 7); + } + + // no feasible solution is better than the front: random objective vectors (the objectives + // are independent, and any vector in the box is reached), every feasible one weakly dominated + // by a point of the front up to its spacing, and dominating none + #[test] + fn random_solutions_agree_with_the_fronts() { + fn check( + front: &[[f64; M]], + values: impl Fn(&[f64; M]) -> Vec, + high: f64, + tolerance: f64, + ) { + let mut rng = StreamRng::seed_from_u64(3); + let real = Real::uniform(M, 0.0..=high).expect("valid bounds"); + let mut feasible = 0; + for _ in 0..20_000 { + let genome = real.random_genome(&mut rng); + let f: [f64; M] = std::array::from_fn(|j| genome[j]); + if values(&f).iter().any(|&g| g > 0.0) { + continue; + } + feasible += 1; + assert!( + front + .iter() + .any(|q| (0..M).all(|j| q[j] <= f[j] + tolerance)), + "{f:?}" + ); + for q in front { + assert!( + !(0..M).all(|j| f[j] < q[j] - 1e-12), + "{f:?} dominates {q:?}" + ); + } + } + assert!(feasible > 1_000, "{feasible}"); + } + check( + &Dtlz8::<3>::front(5_000), + |f| Dtlz8::<3>::values(f).to_vec(), + 1.0, + 0.01, + ); + check( + &Dtlz8::<4>::front(5_000), + |f| Dtlz8::<4>::values(f).to_vec(), + 1.0, + 0.02, + ); + check(&Dtlz9::<3>::front(2_000), Dtlz9::<3>::values, 1.5, 0.01); + check(&Dtlz9::<2>::front(2_000), Dtlz9::<2>::values, 1.5, 0.01); + } + + #[test] + fn sizes() { + assert_eq!(Dtlz8::<5>::default().variables(), 50); + assert_eq!(Dtlz8::<3>::default().constraint_count(), 3); + assert_eq!(Dtlz9::<3>::default().constraint_count(), 2); + assert_eq!(Dtlz9::<2>::new(7).representation().bounds().len(), 7); + } + + #[test] + #[should_panic(expected = "at least 3 objectives")] + fn dtlz8_with_two_objectives_panics() { + let _ = Dtlz8::<2>::default(); + } +} diff --git a/src/multi/problems/mw.rs b/src/multi/problems/mw.rs index 3d39661e..35625d89 100644 --- a/src/multi/problems/mw.rs +++ b/src/multi/problems/mw.rs @@ -243,7 +243,7 @@ fn distance(a: [f64; 2], b: [f64; 2]) -> f64 { // front's extent is a gap between two pieces, and each piece gets points in proportion to its // length and half of the gaps beside it, at least one; within a piece, they're spread evenly by // length, and a piece that is a single point repeats it. Each is a point of the front. -fn spread_by_length(front: &[[f64; 2]], points: usize) -> Vec<[f64; 2]> { +pub(super) fn spread_by_length(front: &[[f64; 2]], points: usize) -> Vec<[f64; 2]> { if points == 0 || front.is_empty() { return Vec::new(); }