Understanding Hypervolume Behavior Theoretically for Benchmarking in Evolutionary Multi/Many-Objective Optimization

摘要

Hypervolume (HV) is one of the most commonly used metrics for evaluating the Pareto front (PF) approximations generated by multiobjective evolutionary algorithms. Even so, HV is a resultant of a complex interplay between the PF shape, number of objectives, and user-specified reference points which, if not well understood, may lead to misinformed inferences about benchmarking performance. In order to understand this behavior, some previous studies have investigated such interactions empirically. In this letter, a new and unconventional approach is taken for gaining further insights about HV behavior. The key idea is to develop theoretical formulas for certain linear (equilateral simplex) and quadratic (orthant) PFs in two specific orientations: 1) regular and 2) inverted. These PFs represent a large number of problems in the existing DTLZ and WFG suites commonly used for benchmarking. The numerical experiments are presented to demonstrate the utility of the proposed work in benchmarking, and in understanding the contributions of the different regions of the PFs, such as corners, edges, as well explaining the contrast between the HV behaviors for regular versus inverted PFs. This letter provides a foundation and computationally fast means to undertake parametric studies to understand various aspects of HV.