Hypervolume-Optimal $\mu$-Distributions on Line/Plane-based Pareto Fronts in Three Dimensions

TL;DR

This paper investigates the distribution of solutions that maximize hypervolume in three-dimensional Pareto fronts, revealing that uniformity is not always optimal and depends on the front's structure.

Contribution

It extends the understanding of hypervolume-optimal distributions to line- and plane-based Pareto fronts in three dimensions, highlighting conditions for uniformity and optimality.

Findings

Solutions are not always uniformly distributed on line-based Pareto fronts.

Uniform solutions on plane-based Pareto fronts are not always hypervolume optimal.

Distribution depends on how multiple lines are combined in the front.

Abstract

Hypervolume is widely used in the evolutionary multi-objective optimization (EMO) field to evaluate the quality of a solution set. For a solution set with $\mu$ solutions on a Pareto front, a larger hypervolume means a better solution set. Investigating the distribution of the solution set with the largest hypervolume is an important topic in EMO, which is the so-called hypervolume optimal $\mu$-distribution. Theoretical results have shown that the $\mu$ solutions are uniformly distributed on a linear Pareto front in two dimensions. However, the $\mu$ solutions are not always uniformly distributed on a single-line Pareto front in three dimensions. They are only uniform when the single-line Pareto front has one constant objective. In this paper, we further investigate the hypervolume optimal $\mu$-distribution in three dimensions. We consider the line- and plane-based Pareto fronts. For the line-based Pareto fronts, we extend the single-line Pareto front to two-line and three-line Pareto fronts, where each line has one constant objective. For the plane-based Pareto fronts, the linear triangular and inverted triangular Pareto fronts are considered. First, we show that the $\mu$ solutions are not always uniformly distributed on the line-based Pareto fronts. The uniformity depends on how the lines are combined. Then, we show that a uniform solution set on the plane-based Pareto front is not always optimal for hypervolume maximization. It is locally optimal with respect to a $(\mu+1)$ selection scheme. Our results can help researchers in the community to better understand and utilize the hypervolume indicator.