Guard optimizers and linesearch against non-descent directions and zero steps

[lkdvos]

A non-descent search direction (e.g. from a preconditioner that is indefinite in floating point, see QuantumKitHub/MPSKit.jl#532) makes the linesearch return a zero step, which no optimizer handled:

• ConjugateGradient hangs: β becomes 0/0 = NaN, the NaN direction passes the descent check, and the bracket expansion loops forever. A zero initial guess (2 * 0) loops there too.
• LBFGS and GradientDescent retry the zero step until maxiter.

fg(x) = (dot(x, x) / 2, copy(x))
precondition(x, g) = [g[1], -g[2] / 2]
optimize(fg, [1.0, 0.1], ConjugateGradient(; gradtol = 1e-12); precondition)  # hangs

Changes

• Optimizers check for descent before the linesearch. CG restarts from the preconditioned gradient on a bad or non-finite β; LBFGS resets its inverse Hessian. They stop with a warning if the preconditioned gradient is not a descent direction or the linesearch makes no progress. CG no longer doubles a zero step.
• HagerZhangLineSearch rejects invalid initial guesses, treats a NaN slope as non-descent, and bounds the bracket expansion by maxfg.

Cost: one extra inner per iteration.

Tests

New tests for non-descent directions (all optimizers), CG with β = NaN or −1e3, and the linesearch guards. They fail or hang on master and pass here.

🤖 Generated with Claude Code

[codecov-commenter]

Codecov Report

❌ Patch coverage is 77.08333% with 11 lines in your changes missing coverage. Please review.
✅ Project coverage is 83.29%. Comparing base (4fc7181) to head (fd8870d).

Files with missing lines	Patch %	Lines
src/lbfgs.jl	64.28%	5 Missing ⚠️
src/cg.jl	87.50%	2 Missing ⚠️
src/gd.jl	71.42%	2 Missing ⚠️
src/linesearches.jl	77.77%	2 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##           master      #48      +/-   ##
==========================================
+ Coverage   82.31%   83.29%   +0.97%     
==========================================
  Files           8        8              
  Lines         752      802      +50     
==========================================
+ Hits          619      668      +49     
- Misses        133      134       +1     

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[lkdvos]

Closes #43.

Partially addresses #44: adds warnings, a CG restart and an LBFGS memory reset. GD and CG no longer get stuck: with a wrong-sign gradient (fg(x) = (x[1]^2, [-1.0])), master hangs and this PR stops after one iteration. LBFGS still runs to maxiter there, because the failing linesearch returns a tiny nonzero step within the ϵ tolerance. Fixing that needs the linesearch to report failure (see #15). The forced-step suggestion is not implemented.

Related, not closed: #15 (only the zero-step case is covered), #11 (bisect is already bounded on master; this PR bounds bracket), #16 (already works on master).

Does not supersede #19. safe_sqrt guards negative inner(x, v, v); this PR guards the sign of inner(x, g, η). The negative inner product in the MPSKit case is fixed at the source in QuantumKitHub/MPSKit.jl#532.

[Jutho]

Nice; great to see some of these infinite loops and other instabilities gone.

[lkdvos]

I don't have any write access here so you'll have to merge/tag this one I fear :)