Multi-Modal Multi-Objective Test Problems with an Infinite Number of Equivalent Pareto Sets

Multi-modal multi-objective optimization problems have multiple equivalent Pareto sets, each of which is mapped to the entire Pareto front. A number of multi-modal multi-objective algorithms have been proposed to find all equivalent Pareto sets. Their performance is evaluated by computational experiments on multi-modal multi-objective test problems. A common feature of those test problems is that a single point on the Pareto front in the objective space corresponds to multiple clearly separated Pareto optimal solutions in the decision space. In this paper, we propose a new type of multi-modal multi-objective test problems where a single point on the Pareto front corresponds to an infinite number of Pareto optimal solutions (i.e., a subset of the decision space). This means that the mapping from the Pareto set in the decision space to the Pareto front in the objective space is a set-to-point mapping. For example, all points on a line in the decision space are mapped to the same single point on the Pareto front. As a result, the dimensionality of the Pareto set is larger than that of the Pareto front. We examine the search behavior of multi-modal multi-objective algorithms using the proposed test problems. Some interesting observations are reported.