Use of Two Reference Points in Hypervolume-Based Evolutionary Multiobjective Optimization Algorithms

Recently it was reported that the location of a reference point has a dominant effect on the optimal distribution of solutions for hypervolume maximization when multiobjective problems have inverted triangular Pareto fronts. This implies that the use of an appropriate reference point is indispensable when hypervolume-based EMO (evolutionary multiobjective optimization) algorithms are applied to such a problem. However, its appropriate reference point specification is difficult since it depends on various factors such as the shape of the Pareto front (e.g., triangular, inverted triangular), its curvature property (e.g., linear, convex, concave), the population size, and the number of objectives. To avoid this difficulty, we propose an idea of using two reference points: one is the nadir point, and the other is a point far away from the Pareto front. In this paper, first we demonstrate that the effect of the reference point is strongly problem-dependent. Next we propose an idea of using two reference points and its simple implementation. Then we examine the effectiveness of the proposed idea by comparing two hypervolume-based EMO algorithms: one with a single reference point and the other with two reference points.