Empirical Hypervolume Optimal µ-Distributions on Complex Pareto Fronts

Hypervolume optimal µ-distribution is the distribution of µ solutions maximizing the hypervolume indicator of µ solutions on a specific Pareto front. Most studies have focused on simple Pareto fronts such as triangular and inverted triangular Pareto fronts. There is almost no study which focuses on complex Pareto fronts such as disconnected and partially degenerate Pareto fronts. However, most real-world multi-objective optimization problems have such a complex Pareto front. Thus, it is of great practical significance to study the hypervolume optimal µ-distribution on the complex Pareto fronts. In this paper, we study this issue by empirically showing the hypervolume optimal µ-distributions on the Pareto fronts of some representative artificial and real-world test problems. Our results show that, in general, maximizing the hypervolume indicator does not lead to uniformly distributed solution sets on the complex Pareto fronts. We also give some suggestions related to the use of the hypervolume indicator for performance evaluation of evolutionary multi-objective optimization algorithms.

I. Introduction

The hypervolume indicator [1] is a well-known performance indicator in evolutionary multi-objective optimization (EMO). It has been widely-used for performance evaluation of EMO algorithms since it can evaluate their convergence and diversity performance simultaneously [2]. The ability to evaluate the convergence performance is due to its Pareto compliance property [3]. Better solution sets with respect to the Pareto dominance relation always have larger hypervolume values. The ability to evaluate the diversity performance is based on the following commonly-believed implicit assumption: A larger hypervolume value means a more diverse solution set. However, this issue is under-investigated in the EMO community.