第16卷第4期 智能系统学报 Vol.16 No.4 2021年7月 CAAI Transactions on Intelligent Systems Jul.2021 D0:10.11992/tis.202010026 网络出版地址：https:/ns.cnki.net/kcms/detail/23.1538.TP.20210407.1558.007html 融合分区和局部搜索的多模态多目标优化 胡洁，范勤勤2，王直欢 (1.上海海事大学物流研究中心，上海201306,2.上海交通大学系统控制与信息处理教有部重点实验室，上海 200240) 摘要：为解决多模态多目标优化中种群多样性维持难和所得等价解数量不足问题，基于分区搜索和局部搜 索，本研究提出一种融合分区和局部搜索的多模态多目标粒子群算法(multimodal multi-.objective particle swarm optimization combing zoning search and local search,ZLS-SMPSO-MM)。在所提算法中，整个搜索空间被分割成多 个子空间以维持种群多样性和降低搜索难度：然后，使用已有的自组织多模态多目标粒子群算法在每个子空间 搜索等价解和挖掘邻域信息，并利用局部搜索能力较强的协方差矩阵自适应算法对有潜力的区域进行精细搜 索。通过14个多模态多目标优化问题测试，并与其他5种知名算法进行比较；实验结果表明ZLS-SMPSO MM在决策空间能够找到更多的等价解，且整体性能要好于所比较算法。 关键词：多模态多目标优化；分区搜索；局部搜索：协方差矩阵自适应策略；种群多样性：等价解：多模态多目标 粒子群算法 中图分类号：TP301.6文献标志码：A文章编号：1673-4785(2021)04-0774-1】 中文引用格式：胡洁，范勤勤，王直欢.融合分区和局部搜索的多模态多目标优化.智能系统学报，2021,16(4)：774-784. 英文引用格式：HU Jie,.FAN Qingin,.WANG Zhihuan..Multimodal multi-.objective optimization combining zoning and local search[JI.CAAI transactions on intelligent systems,2021,16(4):774-784. Multimodal multi-objective optimization combining zoning and local search HU Jie',FAN Qinqin2,WANG Zhihuan' (1.Logistics Research Center,Shanghai Maritime University,Shanghai 201306,China;2.Key Laboratory of System Control and In- formation Processing,Ministry of Education of China,Shanghai JiaoTong University,Shanghai 200240,China) Abstract:To maintain population diversity and find a sufficient number of equivalent solutions in multimodal multi-ob- jective optimization,a multimodal multi-objective particle swarm optimization algorithm with zoning and local searches (ZLS-SMPSO-MM)is proposed in this study.In the proposed algorithm,which is based on zoning search and local search,the entire search space is divided into several subspaces to maintain population diversity and reduce search diffi- culty.Subsequently,an existing self-organizing multimodal multi-objective particle swarm algorithm is used to search equivalent solutions and mine neighborhood information in each subspace,and the covariance matrix adaptation al- gorithm,which has a better local search ability,is utilized for a refined search in promising regions.Lastly,the perform- ance of ZLS-SMPSO-MM is tested on 14 multimodal multi-objective optimization problems and compared with that of other five state-of-the-art algorithms.Experimental results show that the proposed algorithm can find more equivalent solutions in the decision space and its overall performance is better than that of the compared algorithms. Keywords:multimodal multi-objective optimization;zoning search;local search;covariance matrix adaptation evolu- tionary strategy;population diversity;equivalent solutions;multimodal multi-objective particle swarm optimization 收稿日期：2020-10-23.网络出版日期：202104-07. 基金项目：国家重点研发计划项目(20I6YFC0800200):国家自 现实生活中的问题往往会涉及多个优化目 然科学基金项目(61603244)：中国博士后科学基金 项目(2018M642017). 标，且它们可能彼此冲突、相互制约，这类问题被 通信作者：范勤勤.E-mail:foreverl23fan@l63.com 称为多目标优化问题(multi--objective optimizationDOI: 10.11992/tis.202010026 网络出版地址: https://kns.cnki.net/kcms/detail/23.1538.TP.20210407.1558.007.html 融合分区和局部搜索的多模态多目标优化 胡洁1 ，范勤勤1,2，王直欢1 （1. 上海海事大学 物流研究中心，上海 201306; 2. 上海交通大学 系统控制与信息处理教育部重点实验室，上海 200240） 摘 要：为解决多模态多目标优化中种群多样性维持难和所得等价解数量不足问题，基于分区搜索和局部搜 索，本研究提出一种融合分区和局部搜索的多模态多目标粒子群算法 (multimodal multi-objective particle swarm optimization combing zoning search and local search，ZLS-SMPSO-MM)。在所提算法中，整个搜索空间被分割成多 个子空间以维持种群多样性和降低搜索难度；然后，使用已有的自组织多模态多目标粒子群算法在每个子空间 搜索等价解和挖掘邻域信息，并利用局部搜索能力较强的协方差矩阵自适应算法对有潜力的区域进行精细搜 索。通过 14 个多模态多目标优化问题测试，并与其他 5 种知名算法进行比较；实验结果表明 ZLS-SMPSO￾MM 在决策空间能够找到更多的等价解，且整体性能要好于所比较算法。 关键词：多模态多目标优化；分区搜索；局部搜索；协方差矩阵自适应策略；种群多样性；等价解；多模态多目标 粒子群算法 中图分类号：TP301.6 文献标志码：A 文章编号：1673−4785(2021)04−0774−11 中文引用格式：胡洁, 范勤勤, 王直欢. 融合分区和局部搜索的多模态多目标优化 [J]. 智能系统学报, 2021, 16(4): 774–784. 英文引用格式：HU Jie, FAN Qinqin, WANG Zhihuan. Multimodal multi-objective optimization combining zoning and local search[J]. CAAI transactions on intelligent systems, 2021, 16(4): 774–784. Multimodal multi-objective optimization combining zoning and local search HU Jie1 ，FAN Qinqin1,2 ，WANG Zhihuan1 (1. Logistics Research Center, Shanghai Maritime University, Shanghai 201306, China; 2. Key Laboratory of System Control and In￾formation Processing, Ministry of Education of China, Shanghai JiaoTong University, Shanghai 200240, China) Abstract: To maintain population diversity and find a sufficient number of equivalent solutions in multimodal multi-ob￾jective optimization, a multimodal multi-objective particle swarm optimization algorithm with zoning and local searches (ZLS-SMPSO-MM) is proposed in this study. In the proposed algorithm, which is based on zoning search and local search, the entire search space is divided into several subspaces to maintain population diversity and reduce search diffi￾culty. Subsequently, an existing self-organizing multimodal multi-objective particle swarm algorithm is used to search equivalent solutions and mine neighborhood information in each subspace, and the covariance matrix adaptation al￾gorithm, which has a better local search ability, is utilized for a refined search in promising regions. Lastly, the perform￾ance of ZLS-SMPSO-MM is tested on 14 multimodal multi-objective optimization problems and compared with that of other five state-of-the-art algorithms. Experimental results show that the proposed algorithm can find more equivalent solutions in the decision space and its overall performance is better than that of the compared algorithms. Keywords: multimodal multi-objective optimization; zoning search; local search; covariance matrix adaptation evolu￾tionary strategy; population diversity; equivalent solutions; multimodal multi-objective particle swarm optimization 现实生活中的问题往往会涉及多个优化目 标，且它们可能彼此冲突、相互制约，这类问题被 称为多目标优化问题 (multi-objective optimization 收稿日期：2020−10−23. 网络出版日期：2021−04−07. 基金项目：国家重点研发计划项目 (2016YFC0800200)；国家自 然科学基金项目 (61603244)；中国博士后科学基金 项目 (2018M642017). 通信作者：范勤勤. E-mail：forever123fan@163.com. 第 16 卷第 4 期 智 能 系 统 学 报 Vol.16 No.4 2021 年 7 月 CAAI Transactions on Intelligent Systems Jul. 2021