Scalable Multi-Modal Multi-Objective Test Problems with Respect to Decision Space Dimensionality and Pareto Set Dimensionality: High-Dimensional Manhattan Distance Minimization Problems

In most multi-objective test problems, the dimensionality of the Pareto set is the same as the dimensionality of the Pareto front. That is, an m-objective test problem with n decision variables usually has an (m-1)-dimensional Pareto front and an (m-1)-dimensional Pareto set. Thanks to this property, the mapping between the Pareto front and the Pareto set is usually a one-to-one mapping. In this paper, we formulate two-objective distance minimization problems in an n-dimensional decision space using Manhattan distance. The two objectives are defined by two target points in the decision space. That is, each objective is to minimize the Manhattan distance from the solution to each target point. The main characteristic feature of our test problems is that the dimensionality of the Pareto set can be arbitrarily specified between 1 and n by the locations of the two target points. For example, the Pareto set dimensionality in a 20-dimensional decision space is an arbitrarily specified integer between 1 and 20. These test problems have multimodality since many points in the Pareto set are mapped to the same single point on the Pareto front. In this paper, we first explain the relation between the locations of the two target points and the Pareto set dimensionality. Then, through computational experiments, we show that our simple test problems pose difficult challenges for both multi-objective algorithms and multi-modal multi-objective algorithms. We also discuss the scalability of the proposed test problems with respect to the number of equivalent Pareto sets and the number of objectives.