fix: robustify MOSEGO constraint relaxation and expand convergence tests

[Paul-Saves]

This Pull Request finalizes and robustifies the MOSEGO acquisition strategy for multi-objective constrained optimization. It focuses on mathematically correcting the constraint relaxation, enforcing strict seeding compliance, expanding the test suite, and enhancing the notebooks.

• Constraint Relaxation Refactoring
• The relax_constraints parameter has been upgraded from a rigid boolean to a continuous float (default 0.0 ). It now algebraically controls the variance margins applied to constraints (μ ± r × σ).
• Streamlined build_scipy_constraints logic by removing obsolete conditionals for inequality bounds, making the code mathematically precise.
• Extensive Convergence & Unit Testing
• Added hypervolume growth convergence tests for ZDT1 using both EHVI and MPI .
• Added robust constrained convergence tests for BNH ensuring MOSEGO performs flawlessly with both dynamic relaxation ( relax_constraints=2.0 ) and hard constraints ( 0.0 ).
• Substantial unit test coverage added for all MO benchmarks and acquisition functionalities.
• Notebook Enhancements ( multi_obj.ipynb )
• Added visual scatter plots comparing the effects of relax_constraints=2.0 vs 0.0 on the BNH problem (highlighting feasible/infeasible regions).
• Added Pareto front comparisons between EHVI and MPI .

[✓] Code successfully passes ruff check on both src/ and src/tests/
[✓] All Pytest convergence and unit tests pass
[✓] Example notebooks are fully functional and clean

[coveralls]

Coverage Report for CI Build 29764570155

Coverage increased (+0.3%) to 87.913%

Details

• Coverage increased (+0.3%) from the base build.
• Patch coverage: 4 uncovered changes across 4 files (193 of 197 lines covered, 97.97%).
• 3 coverage regressions across 1 file.

Uncovered Changes

File	Changed	Covered	%
src/tests/test_unit/test_ehvi.py	26	25	96.15%
src/tests/test_unit/test_mo_benchmarks.py	43	42	97.67%
src/tests/test_unit/test_mpi.py	33	32	96.97%
src/tests/test_unit/test_zdt.py	38	37	97.37%
Total (12 files)	197	193	97.97%

Coverage Regressions

3 previously-covered lines in 1 file lost coverage.

File	Lines Losing Coverage	Coverage
src/smt_optim/subsolvers/multistart.py	3	47.58%

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Coverage Stats

Relevant Lines:	5204
Covered Lines:	4575
Line Coverage:	87.91%
Coverage Strength:	1.76 hits per line

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💛 - Coveralls

[oihanc]

Although I believe checking for negative variance before computing the square root would be more numerically stable, theoretically, one objective variance could be less than 1e-12 while the other is not. In that case, the EHVI value would not necessarily be equal to 0.
For example, considering the following values:

Y = np.array([
    [3, 1],
    [2, 1.5].
    [1, 2.5],
])

mu = np.array([2, 1.5])
s = np.array([0.7, 1e-12])

The EHVI value is 0.315, even though s is below 1e-6 (sqrt(1e-12)). Furthermore, variance values below 1e-12 are common in the MF context. I do find the limit quite high.

[oihanc]

Since the CDF is multiplied here, it is indeed possible to skip the MPI calculation, when the variance is too low. On that note, I just realized that the return condition when the variance is null should be 0, not 1 (error was already present prior to this PR). However, I still believe that the tolerance of 1e-12 is too high, especially in the MF context.

[oihanc]

There are already very similar tests in tests/test_integration/test_mosego for:

• test_zdt1_mosego_ehvi_hypervolume_growth

• test_dtlz5_mosego_multifidelity_constrained

For tests not requiring multi-fidelity, I would recommend using SmtGPX, since it is much faster (except if the purpose was to test MO-SEGO in the mono-fidelity context with the KRG model).

For test_dtlz5_mosego_multifidelity_constrained, the assert condition does not verify the same condition as mentioned in the comment, as the export_as_dict class method returns sample data from every fidelity level.

[oihanc]

If the relax argument is set to 0, I think we should avoid computing the variance as to limit the computation cost. (when dealing with lower or upper value constraints).

[Paul-Saves]

Thank you so much @oihanc for the work! I've updated the code!

[oihanc]

Hi Paul,

Thanks for the fixes and for continuing to improve MO-SEGO! I had a couple of questions regarding the unit tests.

I'm not sure I fully understand the distinction between test_zdt1_hv.py::test_zdt1_mosego_hv and test_mosego.py::test_no_constraint. As far as I can tell, both tests verify MOSEGO on the ZDT1 problem. I noticed that test_zdt1_mosego_hv also asserts that the dataset is populated and that the hypervolume is greater than 0. Would it make sense to add those assertions to the existing test instead of having a separate one? Or is there a specific behavior that test_zdt1_mosego_hv is intended to cover?

Also, would it make sense to move test_bnh_mosego_constrained_convergence and test_bnh_mosego_constrained_convergence_no_relax into test_integration/test_mosego.py, since they all cover MOSEGO? It seems like that might make the test organization a bit more consistent, but I'm curious to hear your thoughts.

I had originally envisioned the test_convergence directory as a place for benchmarking the convergence properties of the optimization algorithms implemented in SMT-optim. However, I ended up not pursuing the idea because those tests are relatively expensive to run. That's why I was wondering whether grouping the MOSEGO tests under test_integration/test_mosego.py might be a more consistent organization.

Let me know what you think!

[Paul-Saves]

@oihanc I've reduced the number of tests and moved them to reduce the computational time

[oihanc]

Thanks, @paul, for your contribution and for helping make MO-SEGO more robust!