国家自然科学基金项目（面上项目，重点项目，重大项目）(61922072，61976237, 61876169, 61673404 and 61806179)
1.Zhengzhou University;2.Zhongyuan University of Technolog
The National Natural Science Foundation of China (61922072，61976237, 61876169, 61673404 and 61806179)
随着工业生产和日常生活需求的多样化，单个解决方案已经无法满足生产生活的需求. 多模态优化可以为决策者提供多个可行方案，但是早期对多模态优化的研究局限在单目标优化中. 在多目标优化中，也存在多模态优化问题，这些问题存在多个全局或局部帕累托最优解集，找到这些最优解集具有重大的理论和实际意义. 本文首先介绍了多模态多目标优化问题的特点和求解难点；然后综述了求解此类问题的主要方法，总结了这些方法的优缺点；为了说明求解多模态多目标优化问题的整体流程，文章给出了典型算法的求解过程，总结了此类算法的一般框架；随后介绍了常用的多模态多目标优化标准测试函数集和性能评价指标；最后给出了多模态多目标优化的应用领域及未来研究方向.
As the requirements in our daily lives and industrial production become diverse, a single solution cannot meet their demands anymore. Multimodal optimization can provide multiple feasible solutions. However, most of the previous researches on multimodal optimization focus on single-objective optimization. In fact, there are multimodal problems in multiobjective optimization. These problems have multiple local or global Pareto sets and it is of great theoretical and practical significance to find these Pareto sets. The major contents of this paper are summarized as follows. First, it introduces the features and challenges of multimodal multiobjective problems. Second, a survey on existing multimodal multiobjective optimization algorithms is given. The advantages and disadvantages of these algorithms are summarized. Third, the typical multimodal multiobjective optimization algorithm is introduced to illustrate the overall process of solving this kind of problem. Moreover, the general framework of multimodal multiobjective optimization algorithms is also presented. Fourth, multimodal multiobjective benchmark problems and performance indicators are introduced. At last, the applications and future research works are given.