Multimodal Multi-objective Optimization Using A Density-based One-by-One Update Strategy

For real-world optimization problems, a uniformly and widely distributed Pareto optimal set (PS) in the decision space can provide more choices for decision makers. However, most of multi-objective evolutionary algorithms (MOEAs) only consider convergence and diversity in the objective space, which rarely pay attention to diversity in the decision space. Especially for multimodal multi-objective optimization problems (MMOPs), there may exist multiple distinct PSs corresponding to the same Pareto front (PF). Thus, we propose a novel multimodal multi-objective evolutionary algorithm using a density-based one-by-one update strategy in this paper, which considers diversity in both the objective and decision spaces. In the proposed algorithm, once an offspring is generated during evolution, the most crowded subregion with the largest niche count in the objective space has to be identified again, helpful to maintain diversity in the objective space. Furthermore, the harmonic average distance approach is used to estimate the global density of solutions in the decision space, trying to maintain the population's diversity in the decision space. Our proposed algorithm is compared with several state-of-the-art algorithms on MMOPs. The experimental results demonstrate that our algorithm is capable of preserving promising solutions with even distribution in both of decision space and objective space and also shows the superiority on solving the adopted MMOPs.