Distance-Based Weak Convergence Strategy for Multimodal Multi-Objective Optimization

The difficulty of searching for different Pareto optimal solution sets (PSs) in traditional multimodal multi-objective optimization problems (MMOPs) is relatively similar and can be addressed by multimodal multi-objective evolutionary algorithms (MMOEAs) using convergence-first methods. However, the equivalent PSs of imbalanced MMOPs exhibit different levels of search difficulty, posing a great challenge to classical MMOEAs. This paper presents a two-stage evolutionary algorithm with a distance-based weak convergence strategy (DCEA), which conducts a convergence relaxation process during the environmental selection. Unlike the non-dominated sorting strategy, DCEA determines whether a solution should be eliminated by assessing if the Euclidean distance between this solution and non-dominated solutions in the objective space is smaller than a threshold. The weak convergence strategy effectively preserves potential solutions with relatively poor convergence, which outperforms most convergence-first MMOEAs. Moreover, the proposed proportional crowding distance is capable of nonlinearly amplifying the density information of solutions, which is used in the roulette wheel selection to decide which solutions will be selected as parents. Experimental results indicate that DCEA achieves a superior performance compared to five advanced algorithms in both imbalanced and conventional MMOPs.