Theoretically investigating optimal μ-distributions for the hypervolume indicator: First results for three objectives

Several indicator-based evolutionary multiobjective optimization algorithms have been proposed in the literature. The notion of optimal μ -distributions formalizes the optimization goal of such algorithms: find a set of μ solutions that maximizes the underlying indicator among all sets with μ solutions. In particular for the often used hypervolume indicator, optimal μ-distributions have been theoretically analyzed recently. All those results, however, cope with bi-objective problems only. It is the main goal of this paper to extend some of the results to the 3-objective case. This generalization is shown to be not straight-forward as a solution's hypervolume contribution has not a simple geometric shape anymore in opposition to the bi-objective case where it is always rectangular. In addition, we investigate the influence of the reference point on optimal μ-distributions and prove that also in the 3-objective case situations exist for which the Pareto front's extreme points cannot be guaranteed in optimal μ-distributions.