An evolutionary algorithm based on independently evolving sub-problems for multimodal multi-objective optimization

Multimodal multi-objective problems (MMOPs) arise frequently in the real world, in which multiple Pareto optimal solution (PS) sets correspond to the same objective set. Traditional multi-objective evolutionary algorithms (MOEAs) show poor performance in solving MMOPs due to a lack of diversity maintenance in the decision space. Thus, recently, many multimodal multi-objective evolutionary algorithms (MMEAs) have been proposed. However, for most existing MMEAs, they generally have an over-convergence phenomenon, leading to the deterioration of the diversity of the decision space. To address these issues, this paper proposes a MMEAs based on independently evolving sub-problems. The sub-problem independent evolution method fits the definition of MMOPs because sub-problems form meaningful niches when they converge gradually, ensuring the convergence of the objective space, and enhancing the diversity of the decision space. In the environmental selection phase, we propose a two-stage environmental selection strategy that can guarantee both the convergence of the objective space and the distribution of the decision space. Finally, we refer to the k-nearest neighbor deletion strategy in the decision space to guarantee the distributivity of each equivalent PS. The experimental results show that our algorithm has higher competitive performance than seven other state-of-the-art MMEAs on two series of test functions.