Multimodal multi-objective optimization with multi-stage-based evolutionary algorithm

A key issue in tackling multimodal multi-objective optimization problems (MMOPs) is achieving the balance between objective space diversity and decision space diversity to obtain multiple Pareto sets (PSs) while guaranteeing convergence to the Pareto front (PF). However, most of the existing methods for MMOPs face the following two shortages: (i) they put insufficient emphasis on improving decision space diversity, resulting in missing some PSs or PS segments; and (ii) they lack the utilization of promising historical individuals which may help search the PSs. To alleviate these limitations, this paper proposes a novel multi-stage evolutionary algorithm with two improved optimization strategies. Specifically, the proposed method decomposes solving MMOP into two tasks, i.e., the Exploration task and the Exploitation task. The Exploration task first aims to explore the decision space to detect the multiple PSs, then, the Exploitation task aims to enhance the diversities on both objective and decision spaces (i.e., exploiting the PF and PSs). To better search PSs, historical individuals that are well-distributed in the decision space are stored as the evolutionary experience, and then used to generate offspring individuals. Moreover, a new differential evolution is designed to force crowded individuals to move to sparse and undetected regions on the PSs to enhance the diversity of PSs. Extensive experimental studies compare the proposed method with five state-of-the-art methods tailored for MMOPs on two benchmark test suites. The results demonstrate that the proposed method can outperform others in terms of three performance indicators.