Reference Point Specification for Greedy Hypervolume Subset Selection

Hypervolume subset selection (HSS) aims to select a subset with a fixed size from a candidate solution set so that the hypervolume of the subset is maximized. The greedy HSS (GHSS) is the most efficient way for solving the HSS problem. When we use GHSS, we implicitly assume that well-distributed solutions over the entire Pareto front will be selected. However, the distribution of selected solutions by GHSS has not been studied. In this paper, we investigate this issue by examining selected solution subsets for different reference point specifications in GHSS. First, we show that a sufficiently large reference point is a good choice for GHSS to select a well-distributed subset on a triangular Pareto front. However, it is not easy to properly specify a reference point for an inverted triangular Pareto front. Then, we propose a dynamic reference point specification method for GHSS to select a well-distributed subset for various types of Pareto fronts. Static and dynamic reference point specifications are compared through computational experiments using 3- and 5-objective candidate solution sets from various types of Pareto front shapes. The experimental results demonstrate the effect of different reference point specifications on the subsets selected by GHSS and the usefulness of the dynamic reference point specification for GHSS.