SDP solver using R-HMC and simulated annealing: fixes and improvements

[angeloskorakitis]

Fixes and improvements to eigenvalue problems, simulated annealing, Boltzmann (reflective) HMC.

[angeloskorakitis]

Changes to some methods.

[angeloskorakitis]

Benchmark: Volesti vs SDPA

This PR includes a comparative analysis of Volesti against the SDPA library on several (dense) SDP test instances.

Test Configuration

Important Note: For Volesti these benchmarks use the origin as the initial point without explicitly computing an interior point. Future tests will include proper interior point computation for a more comprehensive comparison.

Results

Library	Instance	Expected Min	Computed Min	Error	Status	Time
Volesti	SDP 20x20	-1.6535e+00	-1.6207e+00	3.28e-02	✅ PASS	0.011s
SDPA	SDP 20x20	-1.6535e+00	-1.6535e+00	1.42e-14	✅ PASS	0.055s
Volesti	SDP 20x100	-6.7497e+00	-6.6897e+00	6.00e-02	✅ PASS	0.201s
SDPA	SDP 20x100	-6.7497e+00	-1.6664e+00	8.75e+00	❌ FAIL	0.442s
Volesti	SDP 50x40	-3.1514e+01	-3.0825e+01	6.89e-01	✅ PASS	0.083s
SDPA	SDP 50x40	-3.1514e+01	-3.1514e+01	4.26e-14	✅ PASS	0.255s
Volesti	SDP 50x200	-7.7195e+00	-7.6349e+00	8.46e-02	✅ PASS	1.639s
SDPA	SDP 50x200	-7.7195e+00	-7.3341e+00	3.85e-01	✅ PASS	1.895s
Volesti	SDP 200x15	-3.1610e+12	-1.2305e+02	3.16e+12	❌ FAIL	0.136s
SDPA	SDP 200x15	-3.1610e+12	-3.1610e+12	1.00e+01	✅ PASS	0.117s

Key Observations

• Performance: Volesti demonstrates competitive performance, often faster than SDPA
• Accuracy: SDPA achieves near-machine-precision results when successful, while Volesti's random walk approach provides good approximations within tolerance
• Reliability: Both solvers exhibit failures on different problem dimensions:
  • SDPA fails on 20x100 (error: 5.08)
  • Volesti fails on 200x15 (error: 3.16e+12) - the random walk struggles with this problem geometry
• Trade-offs: Volesti offers speed advantages with acceptable accuracy, while SDPA provides higher precision at the cost of computation time

Future Steps

• Implement initial point computation for more robust comparisons
• Investigate geometry-dependent behavior of the random walk

[vissarion]

@angeloskorakitis thanks for your changes in Spectra/cmake, could you please also apply the required changes in CI so that the tests are passing?