The Magnitude of Dominated Sets: A Pareto Compliant Indicator Grounded in Metric Geometry

TL;DR

This paper introduces magnitude as a new Pareto-compliant indicator for multiobjective optimization, grounded in metric geometry, offering advantages over hypervolume in boundary detection and computational efficiency.

Contribution

It provides the first theoretical analysis of magnitude as an indicator, including formulas, properties, and comparisons with hypervolume.

Findings

Magnitude favors boundary-including populations.

Magnitude is computationally comparable to hypervolume in low dimensions.

Magnitude can effectively approximate Pareto fronts with boundary points.

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 \(\Theta(n\log n)\) time. These results identify magnitude as a mathematically natural and computationally viable alternative to hypervolume for finite Pareto front approximations.