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.

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