[flyingdoubleG]

Description

GrEA's environmental selection does not implement GrEA's grid ranking. On ZDT1 it ends at IGD ≈ 0.036, against ≈ 0.005 for NSGA-II under the same budget.

What GrEA does (Yang et al., 2013; PlatEMO's GrEA/EnvironmentalSelection.m). The last front is filled one solution at a time. At each step it picks the best remaining solution by:

• grid ranking GR(x) = Σ_k G_k(x), the sum of the grid coordinates;
• ties broken by grid crowding distance GCD, which accumulates max(M − GD, 0) over the solutions already chosen;
• then by the grid coordinate point distance GCPD.

After each choice it adjusts GR to keep the chosen set spread out:

• +(M+2) for solutions in the same cell;
• +M for solutions grid-dominated by the chosen one;
• a distance-based penalty PD = M − GD for its other neighbours, propagated to the solutions those neighbours grid-dominate.

What the implementation did.

• GR was defined as the number of grid dominators plus the number of neighbours, not Σ_k G_k.
• Solutions were removed one by one by max(GR, GCD, GCPD), with different update rules and without the PD penalty.
• The grid cell width was (f_max − f_min)/div instead of (ub − lb)/div with half-cell padding.
• div defaulted to 10 for every number of objectives. GrEA's recommended settings are 45 for two objectives, 15 for three, and so on.
• Mating used a tournament on (rank, GR, GCD) rather than Pareto dominance, then grid dominance, then GCD.

This PR

• ports the last-front selection, the grid setting and the mating selection;
• defaults div to the recommended value for n_objs (45, 15, 10, 9, 9, 8, 8, 10, 12 for 2–10 objectives, as in PlatEMO). An explicit div still overrides it.
• vectorizes the PD propagation using the transitivity of grid dominance: PD(r) is the largest direct penalty among the neighbours that grid-dominate r. The selection is still a loop over the solutions to choose, like the previous peeling loop.

Effect (IGD after 200 generations, pop_size=100, seeds 0–2):

	before (div=10)	before, with the recommended div	after
ZDT1 (n=30)	0.0389 / 0.0385 / 0.0319	0.0097 / 0.0087 / 0.0085	0.0068 / 0.0068 / 0.0067
DTLZ2 (m=3, d=12)	0.0875 / 0.1004 / 0.0927	0.0796 / 0.0690 / 0.0744	0.0637 / 0.0631 / 0.0677

Tests

• test_grea_last_selection_matches_reference compares _last_selection with a direct port of the sequential reference loop, including the order-dependent PD propagation. It covers 30 random cases, half of them with coarse values to create shared cells and ties. The same comparison over 300 random cases (2–4 objectives, up to 40 solutions) gave no mismatches.
• test_grea_default_div checks the per-objective defaults.
• unit_test/algorithms/test_moea_generated.py passes.

Checklist

• I have formatted my Python code with ruff.
• I have good commit messages.
• If adding new algorithms, problems, operators:
  • Added related test cases.
  • Added docstring to explain important parameters.

[read-the-docs-community]

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