🤖 AI Summary
This work addresses a critical gap in multi-objective optimization: the absence of a finite-set quality indicator that simultaneously guarantees strict Pareto compliance and effectively evaluates boundary points. The study introduces, for the first time, the concept of “magnitude” from metric geometry to construct a novel scalar indicator that is strictly Pareto compliant. By integrating coordinate projection with magnitude theory from category theory, the proposed metric exhibits both weak and strict set monotonicity and positively identifies boundary solutions—addressing key limitations of the hypervolume indicator. The associated algorithm achieves Θ(n log n) time complexity in two and three dimensions. Empirical results demonstrate that magnitude favors populations containing boundary points and complete Das–Dennis reference grids, whereas hypervolume tends to prefer densely filled interior configurations.
📝 Abstract
We investigate \emph{magnitude} as a new unary and strictly Pareto-compliant quality indicator for finite approximation sets to the Pareto front in multiobjective optimization. Magnitude originates in enriched category theory and metric geometry, where it is a notion of size or point content for compact metric spaces and a generalization of cardinality. For dominated regions in the \(\ell_1\) box setting, magnitude is close to hypervolume but not identical: it contains the top-dimensional hypervolume term together with positive lower-dimensional projection and boundary contributions.
This paper gives a first theoretical study of magnitude as an indicator. We consider multiobjective maximization with a common anchor point. For dominated sets generated by finite approximation sets, we derive an all-dimensional projection formula, prove weak and strict set monotonicity on finite unions of anchored boxes, and thereby obtain weak and strict Pareto compliance. Unlike hypervolume, magnitude assigns positive value to boundary points sharing one or more coordinates with the anchor point, even when their top-dimensional hypervolume contribution vanishes. We then formulate projected set-gradient methods and compare hypervolume and magnitude on biobjective and three-dimensional simplex examples. Numerically, magnitude favors boundary-including populations and, for suitable cardinalities, complete Das--Dennis grids, whereas hypervolume prefers more interior-filling configurations. Computationally, magnitude reduces to hypervolume on coordinate projections; for fixed dimension this yields the same asymptotic complexity up to a factor \(2^d-1\), and in dimensions two and three \(Θ(n\log n)\) time. These results identify magnitude as a mathematically natural and computationally viable alternative to hypervolume for finite Pareto front approximations.