feat: the constrained DTLZ8 and DTLZ9, the DC-DTLZ and the DAS-CMOP problems, each with its own example

[tachsinatalay-code]

Batch 11 of #260: the advanced constrained multi-objective suites, 17 problems, each with its own example.

The problems

All are in multi::problems::all::<M>() and gx.problems, with their fronts.

• DTLZ8 and DTLZ9 ({Dtlz8, Dtlz9}::<M>::new(n), 10M variables by default), from the TIK-112 report, sections 8.8-8.9, eqs. 28-29. The block sums run from ⌊(j−1)n/M⌋+1 to ⌊jn/M⌋, as the plan's erratum says.
  • DTLZ8 needs at least 3 objectives, since its last constraint takes the smallest sum of two different objectives other than f_M.
  • DTLZ8's front, derived: the line f₁ = … = f_{M−1} = t, f_M = 1 − 4t for t in [0, 1/6], plus part of the plane 2f_M + fᵢ + fⱼ = 1. On that part, one objective s is in [(1−t)/4, (1−2t)/2] and the others are equal. For M ≥ 4 it's a 2-D surface, not the whole hyperplane the plan implied.
  • DTLZ9's front: the curve f₁ = … = f_{M−1} = cos θ, f_M = sin θ. The report prints no mean here, so the objectives are sums.
• DC-DTLZ ({Dc1Dtlz1, Dc1Dtlz3, Dc2Dtlz1, Dc2Dtlz3, Dc3Dtlz1, Dc3Dtlz3}::<M>::new(n), with_parameters(n, a, b)), from Li, Chen, Fu and Yao (2019), the supplement and arXiv.
  • Compared with the authors' C++ code (EMOC), read only, since it has no license: a = 3, and b = 0.5 except DC2-DTLZ1's 0.9.
  • DC3's constraints cover the M−1 position variables plus g. This is the only reading that fits the supplement's table 2 (feasible shares of 3⁻ᵐ); "j = 1…m" would make the whole front infeasible.
  • Where this differs from EMOC: it keeps DTLZ3's g factor of 100, as the supplement says the objectives are C1-DTLZ3's. EMOC has 10.
  • Unresolved in the sources: n isn't given anywhere (assumed M + 4 and M + 9). The supplement's DC1 feasible share (10.1%) fits b = 0.95, against its own text's 0.5; the text is followed.
  • The fronts are derived, and agree with EMOC's sampled fronts within a mean distance of 0.011 both ways.
• DAS-CMOP1-9 (DasCmop1 … DasCmop9), from Fan et al. (2020), arXiv v3, tables 2-3 and figure 6. A Difficulty triplet sets the difficulty: the paper's 16 standard triplets of table 3, or any. The defaults are 30 variables and figure 6's triplet.
  • Ambiguities settled by the authors' Java code (read only): g sums from j = 2, a_k² = 0.3 and b_k² = 1.2 are squares, at ζ = 0 the type II constraint is never active, and at ζ = 1 it is |g − 0.5| ≤ 1e-4.
  • The fronts are sampled, as the paper's are: the first feasible point on each ray α + g(1, …, 1), computed exactly (ellipses and spheres are quadratic in g along a ray), then filtered for non-dominance. With 2 objectives, 20,000 rays plus the ends of the type I intervals; with 3, Das-Dennis points at 288 divisions. Cached per triplet.
  • Agreement with the authors' 144 front files: mean distance under 0.01 for 130 files (theirs to ours) and 140 (ours to theirs). Every exception is a point of theirs that breaks the constraints, the point x₁ = 1 at η = 0.5 (feasible only in exact arithmetic), or a corner region of DAS-CMOP8/9 that their files omit and that is non-dominated.

Examples

17 examples at orders 179-195, the free range between C3-DTLZ4 (178) and convex DTLZ2 (200); no existing example moves. Rust and Python give identical output and identical traces.

Example	Main method	Result
DAS-CMOP4-6	NSGA-II, η 5, the paper's 300 × 1000	IGD+ ≤ 0.01 in 20, 20 and 19 of 20 seeds
DAS-CMOP7-8	NSGA-III, 276 directions, η 5	20/20 at ≥ 99% of a same-size front sample's hypervolume
DC1-DTLZ1	NSGA-III (C-NSGA-III's settings), 1,000 gen.; SMS-EMOA as contrast	20/20
DC1-DTLZ3	NSGA-III, 2,000 gen.	13/20 at the target, the rest within 2%
DC2-DTLZ1/3	NSGA-III on plain DTLZ, then scored by DC2	20/20 and 18/20
DC3-DTLZ1/3	NSGA-III without the constraint on g	18/20 and 20/20
DTLZ8	SMS-EMOA, 20,000 gen.	99.5-100.7% of the sample's hypervolume
DTLZ9	SMS-EMOA, 20,000 gen.	99.8-99.9%, IGD+ 0.0025 (the sample's own)
DAS-CMOP1-3, 9	NSGA-II / NSGA-III	don't reach the front: IGD+ about 0.7, 0.15, 0.16, 0.19

DAS-CMOP1-3 and 9: their distance variables are linked, and reaching the front needs differential-evolution-style variation, as the paper's MOEA/D-CDP has. None of the operators available to genoxide's multi-objective algorithms moves linked variables together. NSGA-II, NSGA-III, MOEA/D and SMS-EMOA were tried with SBX, blend and arithmetic crossover; all fail, as the paper's NSGA-II-CDP does. The READMEs say so. A DE-style variation for the multi-objective algorithms would close this gap.

DC2 and DC3: the main runs drop some constraints and score the result with the real problem, following the C1-DTLZ3 example. Each example also shows constraint dominance alone stalling, as the C-TAEA paper's C-NSGA-III does: no feasible solution on DC2 in 20 of 20 runs, and stuck in a band of g on DC3.

DTLZ8 and DTLZ9: each also shows NSGA-II for the report's 500 generations failing (DTLZ9 IGD+ about 4).

Tests

Locally, rebased on today's main:

• Lint: fmt, clippy with and without features, and cargo doc with and without features, all with -D warnings on Rust 1.99 (WSL).
• Rust: cargo test --all-features passes, including 649 lib tests and the doc tests.
• Python: mypy --strict genoxide reports no issues, and pytest passes (991 passed, 1 skipped).
• Examples: all 17 run in both languages with identical output and traces, and examples/check_traces.py passes for all traces.