A Two-Stage Fitness Learning Model-Driven Evolutionary Algorithm for Imbalanced Multimodal Multi-Objective Optimization

In recent years, multimodal multi-objective optimization problems (MMOPs) have become a hot research topic in the field of evolutionary computation in recent years, whose main goal is to locate all equivalent Pareto-optimal solution sets. Although existing evolutionary multimodal multi-objective algorithms (MMOAs) perform well when there is no obvious difference in the search difficulty of different Pareto-optimal solution sets, they face great challenges when such difficulty differences are prominent, as most current MMOAs fail to effectively address the imbalance of fitness landscapes, leading to an inability to stably find all Pareto-optimal modes and poor robustness in complex MMOPs. To fill this gap, the main objective of this study is to propose a novel MMOA that can adapt to imbalanced fitness landscapes, thereby improving the ability to locate all Pareto-optimal solution sets and enhancing the algorithm’s robustness. To achieve this objective, a novel multimodal multi-objective evolutionary algorithm based on a two-stage fitness learning model is proposed. First, a multi-subpopulation cooperative search strategy is designed. Based on the principle of speciation, this strategy divides the population into several subpopulations, with the formation of each subpopulation guided by individual similarity in the decision space, thereby guiding the population to perform decentralized search across different modes. Second, a two-stage fitness learning model is developed. In the early and middle stages of evolution, individual fitness is evaluated by integrating Pareto dominance strength and density estimates based on the local outlier factor; in the late stage of evolution, individual fitness is evaluated using fast non-dominated sorting and twin-mirror crowding distance. The former is used to balance the convergence and diversity of the population in the decision space, while the latter is used to improve the convergence and diversity of the population in both the decision space and the objective space. Finally, simulation experiments are conducted on 12 imbalanced multimodal multi-objective optimization problems, and the results are compared to those of seven popular evolutionary multimodal multi-objective optimization algorithms. The results demonstrate that the proposed algorithm can find all modes for different problems and exhibits better robustness.