Population Diversity Dynamics Analysis for Imbalanced Multi-objective Optimization
针对进化多目标优化中的非平衡问题，提出了全局和局部收缩扩散率来刻画种群收敛与多样性动态，通过概率分析证明多样性恢复概率低于0.1，并构建了十个可调难度的基准问题，为设计鲁棒算法提供理论指导。
Within the field of evolutionary multi-objective optimization, there exists a class of problems where most evolutionary multi-objective optimization algorithms (EMOAs) suffer from a loss of diversity that remains unrecoverable for extended periods, namely “imbalanced problems”. Currently, the field of multi-objective optimization lacks rigorous quantitative characterization of imbalanced multi-objective optimization problems (MOPs) and systematic understanding of their impact on algorithmic performance. To address these limitations, we introduce two novel quantities: the global and local shrinkage-spread rates, which characterize population convergence and diversity dynamics during global exploration and local exploitation phases, respectively. Based on these quantities, we provide the first mathematical characterization of imbalanced MOPs and derive key theoretical properties through rigorous probabilistic analysis. Our theoretical result shows that the probability of achieving substantial diversity recovery is below 0.1 under given conditions, demonstrating that diversity recovery becomes extremely challenging once population diversity is lost. We further construct ten imbalanced MOPs in which the difficulty of maintaining population diversity can be precisely modulated through adjustable parameters. Using these benchmark problems, we conduct comparative experiments evaluating four representative EMOAs, i.e., NSGA-II, MOEA/D, RVEA, MOEA/D-M2M and DrEA, in terms of their diversity maintenance capabilities on imbalanced MOPs. The results provide both theoretical insights and practical guidance for designing robust EMOAs capable of handling imbalanced MOPs.