Both problem characteristics in multimodality and multi-objective are involved in multimodal multi-objective optimization problems (MMOPs). How to locate diverse Pareto sets and approximate Pareto front simultaneously is a challenging research topic. To address this issue, a cluster-based immune-inspired algorithm using manifold learning is proposed in this paper for solving MMOPs. First of all, the population is partitioned into multiple subpopulations, and each of them is expected to find equivalent Pareto solutions in different regions. Subsequently, the immune-inspired algorithm with proportional cloning and hypermutation is developed for improving the diversity of the population and obtaining high-quality Pareto solutions in the decision space. Additionally, principal component analysis is adopted to learn the manifold of the Pareto set, further improve the convergence, and enhance interaction among subpopulations. The proposed algorithm is compared with six state-of-the-art algorithms. Experimental results demonstrate that the proposed algorithm is capable of locating equivalent Pareto optimal solutions in the decision space and maintaining the diversity and convergence of solutions in both decision space and objective space, simultaneously.

多模态和多目标中的问题特征都涉及多模态多目标优化问题（MMOP）。如何同时定位不同的帕累托集和近似帕累托前沿是一个具有挑战性的研究课题。为了解决这个问题，本文提出了一种使用流形学习的基于集群的免疫启发算法来解决 MMOP。首先，将种群划分为多个子种群，每个子种群都期望在不同的区域找到等效的帕累托解。随后，开发了具有比例克隆和超变异的免疫启发算法，以提高种群的多样性并在决策空间中获得高质量的帕累托解。此外，采用主成分分析来学习帕累托集的流形，进一步提高收敛性，增强亚群之间的互动。将所提出的算法与六种最先进的算法进行比较。实验结果表明，该算法能够在决策空间中定位等效帕累托最优解，并同时保持决策空间和目标空间中解的多样性和收敛性。