Riesz s-Energy as a Diversity Indicator in Evolutionary Multi-Objective Optimization

Measuring the diversity of a Pareto Front Approximation (PFA) is critical when comparing the performance of Multi-Objective 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 paper, we propose the Riesz s-energy (Es) as a QI to evaluate the diversity and spread of PFAs. Theoretical results show that Es holds (1) some of the Weitzman properties of a desirable diversity QI, (2) monotonicity, (3) the submodularity property (for -Es), and (4) that it is invariant under rotations. We provide numerical evidence on the behavior of Es in both artificial PFAs and PFAs generated by state-of-the-art MOEAs. The mathematical properties that Es satisfies show its usefulness when it is utilized as a diversity QI in Evolutionary Multi-Objective Optimization.