Many-objective and many-variable test problems for visual examination of multiobjective search

In the development of evolutionary multiobjective optimization (EMO) algorithms, it is important to implement a good balancing mechanism between the convergence of solutions towards the Pareto front and their diversity over the Pareto front. When an EMO algorithm is applied to a two-objective problem, the balance can be easily visualized by showing all solutions at each generation in the two-dimensional objective space. However, such a visual examination of the multiobjective search is difficult for many-objective problems with four or more objectives. The use of many-objective test problems with two decision variables has been proposed in some studies to visually examine the search behavior of EMO algorithms. Such test problems are defined by a number of points in a two-dimensional decision space where the distance minimization from each point is an objective. Thus the number of objectives is the same as the number of points. The search behavior of EMO algorithms can be visually examined in the two-dimensional decision space. In this paper, we propose the use of many-objective test problems for visual examination of the search behavior in a high-dimensional decision space. More specifically, our m-objective test problem with n variables is generated by specifying m points on a plane in an n-dimensional decision space. We examine the behavior of EMO algorithms through computational experiments on such an m-objective n-variable test problem. Our experimental results show that the number of variables has a large effect on the search behavior of EMO algorithms with respect to the diversity of solutions.