Dimensionality Reduction in Multiobjective Optimization with (Partial) Dominance Structure Preservation: Generalized Minimum Objective Subset Problems

Abstract

Most of the available multiobjective evolutionary algorithms (MOEA) for approximating the Pareto set have been designed for and tested on low dimensional problems (≤ 3 objectives). However, it is known that problems with a high number of objectives cause additional difficulties in terms of the quality of the Pareto set approximation and running time. Furthermore, the decision making process becomes the harder the more objectives are involved. In this context, the question arises whether all objectives are necessary to preserve the problem characteristics. One may also ask under which conditions such an objective reduction is feasible, and how a minimum set of objectives can be computed. In this paper, we propose a general mathematical framework, suited to answer these three questions and corresponding algorithms, exact and heuristic ones. The heuristic variants are geared towards direct integration into the evolutionary search process. Moreover, extensive experiments for four well-known test problems show that substantial dimensionality reductions are possible on the basis of the proposed methodology.