Two-objective solution set optimization to maximize hypervolume and decision space diversity in multiobjective optimization

Diversity maintenance in the decision space is a recent hot topic in the field of evolutionary multiobjective optimization (EMO). In this paper, we propose the use of a decision space diversity measure as an objective function in a two-objective formulation of solution set optimization where the hypervolume measure is used as the other objective. In the proposed approach, a given multiobjective problem with an arbitrary number of objectives is handled as a two-objective solution set optimization problem. A solution of our two-objective problem is a set of non-dominated solutions of the original multiobjective problem. An EMO algorithm is used to search for a number of solution sets along the tradeoff surface between the diversity maximization in the decision space and the hypervolume maximization in the objective space. In this paper, first we numerically examine the diversity measure of Solow & Polasky (1994), which was used in recent studies of Ulrich et al. (2010, 2011), through computational experiments on many-objective distance minimization problems in a two-dimensional decision space. Then we formulate a two-objective solution set optimization problem to maximize the decision space diversity and the objective space hypervolume. Finally we demonstrate that a number of non-dominated solution sets can be obtained along the diversity-hypervolume tradeoff surface. Through computational experiments, we also examine the difference between the following two settings for diversity calculation: All solutions in a solution set are used in one setting while only non-dominated solutions are used in the other setting.