Visual examination of the behavior of EMO algorithms for many-objective optimization with many decision variables

Various evolutionary multiobjective optimization (EMO) algorithms have been proposed in the literature. They have different search mechanisms for increasing the diversity of solutions and improving the convergence to the Pareto front. As a result, each algorithm has different characteristics in its search behavior. Multiobjective search behavior can be visually shown in an objective space for a test problem with two or three objectives. However, such a visual examination is difficult in a high-dimensional objective space for many-objective problems. The use of distance minimization problems has been proposed to examine many-objective search behavior in a two-dimensional decision space. This idea has an inherent limitation: the number of decision variables should be two. In our former study, we formulated a four-objective distance minimization problem with 10, 100, and 1000 decision variables. In this paper, we generalize our former study to many-objective problems with an arbitrary number of objectives and decision variables by proposing an idea of specifying reference points on a plane in a high-dimensional decision space. As test problems for computational experiments, we generate six-objective and eight-objective problems with 10, 100, and 1000 decision variables. Our experimental results on those test problems show that the number of decision variables has large effects on multiobjective search in comparison with the choice of an EMO algorithm and the number of objectives.