Hypervolume Optimal $$\mu $$-Distributions on Line-Based Pareto Fronts in Three Dimensions

Hypervolume optimal \(\mu \)-distribution is a fundamental research topic which investigates the distribution of \(\mu \) solutions on the Pareto front for hypervolume maximization. It has been theoretically shown that the optimal distribution of \(\mu \) solutions on a linear Pareto front in two dimensions is the one with \(\mu \) 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 \(\mu \)-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.