Riesz s-Energy as a Diversity Indicator in Evolutionary Multiobjective Optimization

Abstract

Measuring the diversity of a Pareto front approximation (PFA) is critical when comparing the performance of multiobjective evolutionary algorithms (MOEAs). In the literature, some quality indicators (QIs) measure diversity according to their specific preferences. However, just a few QIs have mathematical properties proven. In this article, we propose the Riesz s-energy $(E_{s})$ as a QI to evaluate the diversity and spread of PFAs. Theoretical results show that $E_{s}$ holds 1) some of the Weitzman properties of a desirable diversity QI; 2) monotonicity; 3) the submodularity property (for $-E_{s}$ ); and 4) that it is invariant under rotations. We provide numerical evidence on the behavior of $E_{s}$ in both artificial PFAs and PFAs generated by state-of-the-art MOEAs. The mathematical properties that $E_{s}$ satisfies show its usefulness when it is utilized as a diversity QI in evolutionary multiobjective optimization.