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

Hypervolume is widely used in the evolutionary multiobjective optimization (EMO) field to evaluate the quality of a solution set. For a solution set with
$\mu $
solutions on a Pareto front, a larger hypervolume means a better solution set. Investigating the distribution of the solution set with the largest hypervolume is an important topic in EMO, which is the so-called hypervolume-optimal
$\mu $
-distribution. Theoretical results have shown that the
$\mu $
solutions are uniformly distributed on a linear Pareto front in two dimensions. However, the
$\mu $
solutions are not always uniformly distributed on a single-line Pareto front in three dimensions. They are only uniform when the single-line Pareto front has one constant objective. In this article, we further investigate the hypervolume-optimal
$\mu $
-distribution in three dimensions. We consider the line-based and plane-based Pareto fronts. For the line-based Pareto fronts, we extend the single-line Pareto front to two-line and three-line Pareto fronts, where each line has one constant objective. For the plane-based Pareto fronts, the linear triangular and inverted triangular Pareto fronts are considered. First, we show that the
$\mu $
solutions are not always uniformly distributed on the line-based Pareto fronts. The uniformity depends on how the lines are combined. Then, we show that a uniform solution set on the plane-based Pareto front is not always optimal for hypervolume maximization. It is locally optimal with respect to a
$(\mu +1)$
selection scheme. Our results can help researchers in the community to better understand and utilize the hypervolume indicator.