Optimal Distributions Of Solutions For Hypervolume Maximization On Triangular And Inverted Triangular Pareto Fronts Of Four-Objective Problems

Hypervolume (HV) has been often employed for the comparison of evolutionary multiobjective optimization algorithms. In HV-based performance comparison, it is implicitly assumed that a uniformly distributed solution set on the whole Pareto front has a larger HV than a partially distributed solution set on a part of the Pareto front. In this study, we demonstrate that this assumption does not hold for multiobjective problems whose Pareto fronts are not triangular. When we use a reference point close to the nadir point for an inverted triangular Pareto front, the optimal solution set includes no solutions on the boundary of the Pareto front. However, a set of only boundary solutions (with no inside solutions) is optimal when the reference point is very far away from the nadir point. For a multiobjective problem with four objectives, we explain that such a biased distribution of solutions toward the boundary of the Pareto front looks a good solution set in a parallel coordinate plot. That is, misleading conclusions can be obtained even when we use both the HV-based evaluation and the parallel coordinate-based examination. Our observations in this paper suggest the necessity of projection-based examination of solutions for many-objective problems.