Extended Results on Analytical Hypervolume Indicator Calculation of Linear and Quadratic Pareto Fronts

Hemant Kumar Singh

Abstract: In evolutionary multi-objective optimization (EMO), the quality of Pareto front (PF) approximations obtained using various algorithms are benchmarked using set-based unary indicators. Hypervolume (HV) is an indicator that has attracted widespread attention as it attempts to capture both convergence and diversity, is scalable in terms of objectives, and is Pareto-compliant. Even so, HV is not without its limitations, ignoring which may lead to erroneous judgments about algorithm performance, especially for problems with more than three objectives, where visualization of the objective space is not straightforward. Various theoretical studies have therefore been conducted to improve understanding of HV behavior and its implications in EMO benchmarking, including a recent one that focuses on computing HV for some common PF shapes analytically. This paper aims to extend these theoretical results and discussions, by analyzing continuous subsets (symmetric patches) on some common shapes of PFs (linear and quadratic). Towards this end, a simple parametrization scheme is proposed to represent the symmetric patches on linear and quadratic PFs, and their HV is analytically derived. The resulting expressions are used to observe some HV trends with respect to the patch size, reference point specification and number of objectives, to provide related insights for benchmarking.

External IDs:dblp:conf/emo/Singh25