An analysis of the Hypervolume Sharpe-Ratio Indicator

Abstract

Set-quality indicators have been used in Evolutionary Multiobjective Optimization Algorithms (EMOAs) to guide the search process. Recently, a new class of set-quality indicators combining the selection of solutions with fitness assignment has been proposed. This class is based on a formulation of fitness assignment as a Portfolio Selection Problem, where solutions are seen as assets whose returns are random variables, and fitness represents the investment in such assets/solutions. The Hypervolume Sharpe Ratio (HSR) Indicator is an instance of this class of indicators which has led to promising results as part of an EMOA denominated the Portfolio Optimization Selection Evolutionary Algorithm (POSEA). In this paper, the class of Sharpe-Ratio Indicators is formalized, and the HSR indicator is studied in regard to monotonicity, sensitivity to objective scaling, and dependence on its parameters. In addition, optimal μ-distributions on two-objective linear fronts, and the corresponding fitness assignments, are characterized. Such optimal μ-distributions turn out to be identical to those of the Hypervolume Indicator on the same fronts. Experimental results complement the analysis.