Bi-Directional Coevolution for Multimodal Multiobjective Optimization With Local Pareto Sets

Multimodal multiobjective optimization problems (MMOPs) feature multiple Pareto sets (PSs) corresponding to a single Pareto front (PF). While most multimodal multiobjective evolutionary algorithms (MMEAs) focus on locating multiple global PSs, they tend to overlook local PSs that provide valuable decision diversity despite exhibiting slightly inferior objective values. In this paper, a bi-directional coevolutionary algorithm, termed MMEA-BDC, is proposed for MMOPs with local PSs, which integrates tractive and diffusive search mechanisms to simultaneously discover both global and local PSs. Specifically, the tractive mechanism preserves the individuals with the best fitness, which tend to be located around global PSs, thereby guiding the search toward the global PF. Meanwhile, the diffusive mechanism evaluates individual fitness based on an improved local convergence indicator, allowing individuals near local PSs to be identified and preserved, thereby promoting convergence toward local PF. These individuals are further refined through a niche-driven filter to retain those contributing to decision space diversity. Additionally, a dual-space neighbor replacement strategy is designed to comprehensively consider the crowding degree of individuals in both decision and objective spaces, effectively balancing population diversity across the two spaces. Experimental results on several benchmark suites of MMOPs demonstrate that MMEA-BDC effectively discovers both global and local PSs and achieves superior competitiveness compared to seven state-of-the-art MMEAs.