Balancing Convergence and Diversity in Objective and Decision Spaces for Multimodal Multi-Objective Optimization

Solving multimodal multi-objective optimization problems (MMOPs) via evolutionary algorithms receives increasing attention recently. Maintaining good diversity in both decision and objective spaces is essential to handling MMOPs. Unfortunately, most of the existing methods prefer convergence in the objective space, resulting in the elimination of poorly converged solutions that may be helpful to improve the diversity in the decision space. To overcome this drawback, we propose a coevolutionary algorithm to balance the convergence and the diversity in both objective and decision spaces to better solve MMOPs. In the proposed method, a convergence-first population aims at pursuing a solution set well distributed on both the Pareto front and Pareto set assisted by a convergence-relaxed population. Further, a novel objective relaxation technique is developed for the convergence–relaxed population, which can supplement Pareto set segments not detected by the convergence-first population. Additionally, the environmental selections, mating selection, and fitness evaluation strategies are customized to bring about the balance of convergence and diversity in both objective and decision spaces. Experimental studies on four MMOP benchmarks demonstrated the superiority of the proposed algorithm over six state-of-the-art methods tailored for MMOPs.