Many-objective test problems with multiple Pareto optimal regions in a decision space

In evolutionary multi-objective optimization (EMO) algorithms, diversity maintenance has been mainly discussed in the objective space in order to search for uniformly distributed non-dominated solutions along the entire Pareto front. In this paper, we propose three types of many-objective test problems with multiple Pareto optimal regions in the decision space. One type has multiple equivalent Pareto optimal regions. Another type has different but somewhat similar Pareto optimal regions. The other type has Pareto and local Pareto optimal regions. Our many-objective test problems are generated by placing multiple polygons of the same or similar shapes in a decision space. The ith objective is the minimization of the distance from a solution to the nearest ith vertex over all polygons. Thus the number of objectives is the same as the number of vertices of the polygons. The number of equivalent or similar Pareto regions in the decision space is the same as the number of the polygons.