Hypervolume Subset Selection for Triangular and Inverted Triangular Pareto Fronts of Three-Objective Problems

Hypervolume subset selection is to find a pre-specified number of solutions for hypervolume maximization. The optimal distribution of solutions on the Pareto front has been theoretically studied for two-objective problems in the literature. In this paper, we discuss hypervolume subset selection for three-objective problems with triangular and inverted triangular Pareto fronts. Our contribution is to show that the effect of the location of a reference point for hypervolume calculation on the optimal distribution of solutions is totally different between triangular and inverted triangular Pareto fronts. When the reference point is far from the Pareto front, most solutions are on the sides of the inverted triangular Pareto front while they are evenly distributed over the entire triangular Pareto front. These properties seem to hold in multiobjective problems with four or more objectives. We also show that the effect of the location of a reference point on the optimal distribution is totally different between maximization and minimization problems with the same triangular Pareto fronts. This property is supported by the fact that maximization problems with triangular Pareto fronts are equivalent to minimization problems with inverted triangular Pareto fronts. The optimal distribution of solutions is also discussed when the reference point is close to the Pareto front (i.e., when its location is between the nadir point and the Pareto front).

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