A ring-hierarchy-based evolutionary algorithm for multimodal multi-objective optimization

Multimodal multi-objective optimization problems (MMOPs) involve multiple equivalent Pareto sets (PSs) with identical Pareto front (PF). Popular multimodal multi-objective evolutionary algorithms (MMEAs) are capable of finding multiple equivalent PSs. However, most of MMEAs lead to imbalanced or local PSs are dominated and lost when tackling several MMOPs with the imbalance between convergence and diversity (MMOP-ICD) or MMOP with local Pareto solutions (MMOPL). To tackle this issue, we propose a ring-hierarchy-based evolutionary algorithm for multimodal multi-objective optimization. A ring-based niche technique is used based on the Pareto-based ranking hierarchy. Each hierarchy and its upper and lower neighbors hierarchy form a ringhierarchy topology structure. Subsequently, a local convergence quality that considers the dominance relationship and objective values between all individuals is involved in the ring-hierarchy-based evolutionary strategy. It updates individuals and improves the population convergence quality. Moreover, a distance-based dominance selection that considers the distance between the neighbors and the dominance relationship is also developed. In this case, some individuals that approach imbalanced PS and local PS are maintained in the population instead of being dominated. Meanwhile, a dual-crowding distance is also involved in distance-based dominance selection to select diverse individuals. The proposed algorithm and several state-of-the-art MMEAs are tested on several MMOPs benchmarks. The experimental results demonstrate that the proposed algorithm is competitive and is capable of locating imbalanced PSs and local PSs.