Closure-Guided Optimization: Minimum Structural Repair as a General Constraint-Handling Principle

Constraint handling in population-based optimization is usually organized around violation magnitudes, feasibility rules, penalties, or repair operators, but these mechanisms do not treat the minimum admissible cost of reaching feasibility as the primitive search quantity. Building on Closure Complexity, this paper develops Closure-Guided Optimization (CGO) and Feasibility Closure Complexity (FCC), defined as the least admissible repair cost from a candidate to the feasible locus. We establish a universal repair lower bound, positive-rescaling invariance, an optimality criterion based on the repair ratio, and ranking stability for approximate FCC. The framework is evaluated across genetic algorithms, differential evolution, particle swarm optimization, a separable CMA-ES host, NSGA-II, and ant-colony optimization. Five experimental blocks examine controlled representation invariance; external constrained problems, including welded-beam engineering design; disconnected feasible regions with learned FCC; constrained multiobjective search; and exact interaction-aware set-cover repair, supplemented by an ACO discrete-host validation. Across 4,770 continuous optimization runs, 4,765 repair states, and 60 ACO runs, FCC is most useful when violation ranking diverges from repair difficulty, while normalized violation matches FCC when both encode the same geometry. Interaction-aware joint repair reaches a median cost ratio of 1.000 relative to exact FCC, with an 18.0% mean cost saving, whereas mixed NSGA-II results and the ACO null case show that FCC does not provide a universal advantage. These results support CGO as a conditional, measurable constraint-handling principle.