Two-stage multimodal multi-objective evolutionary algorithm based on global density updating strategy

Multimodal multi-objective problems involve identifying global and local optimal Pareto solution sets. Providing decision-makers with comprehensive Pareto sets is crucial for theoretical understanding and practical application. In recent years, numerous evolutionary algorithms have been proposed to address these issues. However, most of these algorithms first select individuals based on their convergence properties to form mating pools and then consider individuals with good diversity in decision and objective spaces. This method often leads to the replacement of individuals with characteristics in the decision space by those who exhibit strong convergence in the objective area. Additionally, achieving uniform convergence of individuals to each Pareto subregion is challenging because of the varying shapes and positions of Pareto optimal subsets in the decision space.In this paper, we introduce a novel two-stage multimodal multi-objective evolutionary algorithm that employs a global density updating strategy. First, we propose an alternative two-stage optimization framework to address the limitations of traditional mating pools and one-to-one parent-progeny comparisons. The proposed framework dynamically adjusts the population size at different stages and uses distinct optimization methods, promoting a balanced exploration and exploitation of the population. Second, to resolve the uneven distribution of Pareto subsets, we designed adaptive and global density estimation strategies that update the objective space density and population. These mechanisms ensure the maintenance of diversity within the objective and decision spaces. We compared the proposed algorithm with seven representative multimodal multi-objective algorithms. The experimental results demonstrate that the proposed algorithm finds equivalent solutions in the decision space and effectively balances diversity and convergence between the decision and objective space. Overall, the proposed algorithm outperforms the compared algorithms.