Hypervolume Optimal μ-Distributions on Line-Based Pareto Fronts in Three Dimensions

摘要

Hypervolume optimal μ-distribution is a fundamental research topic which investigates the distribution of μ solutions on the Pareto front for hypervolume maximization. It has been theoretically shown that the optimal distribution of μ solutions on a linear Pareto front in two dimensions is the one with μ equispaced solutions. However, the equispaced property of an optimal distribution does not always hold for a single-line Pareto front in three dimensions. It only holds for the single-line Pareto front where one objective of the Pareto front is constant. In this paper, we further theoretically investigate the hypervolume optimal /it-distribution on line-based Pareto fronts in three dimensions. In addition to a single-line Pareto front, we consider Pareto fronts constructed with two lines and three lines, where each line is a Pareto front with one constant objective. We show that even the equispaced property holds for each single-line Pareto front, it does not always hold for the Pareto fronts combined with them. Specifically, whether this property holds or not depends on how the lines are combined.