Solving Multimodal Multiobjective Problems Through Zoning Search

Abstract

Finding a good Pareto front (PF) approximation and locating sufficient equivalent Pareto optimal solutions are two important goals of the multimodal multiobjective optimization (MMO). Preserving the diversity in decision and objective spaces is a core task in the MMO accordingly. Although various “soft isolation” approaches, such as niching methods, have been proposed to promote the diversity and find multiple Pareto optimal solutions in the decision space, they may perform poorly on complex MMO problems (MMOPs) due to high environmental selection pressure and complex geometry of Pareto optimal sets (PSs). To alleviate the above-mentioned challenging task, a “hard/physical isolation” method called zoning search (ZS) is proposed to maintain the diversity in the decision space and reduce the problem complexity in this article. In the ZS, some decision variables of MMOPs are selected randomly and then divided into several segments, i.e., the entire search space is partitioned into many subspaces. Clearly, the population diversity can be naturally maintained in the decision space and the problem complexity is reduced by the ZS in each subspace. The effectiveness of the ZS is systematically evaluated by 11 recently proposed MMOPs. The experimental results demonstrate that the ZS can effectively assist a selected multimodal multiobjective evolutionary algorithm (MMOEA) in finding more and better distributed equivalent Pareto optimal solutions in the decision space, and keep its performance in the objective space unchanged. Additionally, if additional computational resources are given, the ZS can further help the selected MMOEA to improve its performance in the decision space when compared with a soft isolation method used in the corresponding MMOEA. Overall, the ZS is a simple and promising approach to balance the broad search and the deep search in solving MMOPs.