第4期 胡洁，等：融合分区和局部搜索的多模态多目标优化 ·783· 小，这将直接影响算法的搜索效率。同时，当算 ing multimodal multi-objective problems[J].IEEE transac- 法种群规模在400时，算法的性能最佳。但为与 tions on evolutionary computation,2018,22(5):805-817. 第3.2节中比较算法的种群规模保持一致，所提 [5]LIANG Jing,GUO Qiangian,YUE Caitong,et al.A self- 算法的种群规模仍设置为800。 organizing multi-objective particle swarm optimization al- gorithm for multimodal multi-objective problems[C]//Pro- 35 3.2000 ceedings of the 9th International Conference on Swarm In- 3.0 telligence.Shanghai,China:Springer,2018:550-560. 2.7333 [6]LI Zhihui,SHI Li,YUE Caitong,et al.Differential evolu- 2.5 tion based on reinforcement learning with fitness ranking 2.0000 2.0667 2.0 for solving multimodal multiobjective problems[J].Swarm and evolutionary computation,2019,49:234-244. 1.5 [7]FAN Qing,YAN Xuefeng.Solving multimodal multiob- 1.0 jective problems through zoning search[J].IEEE transac- 400 800 12001600 种群规模 tions on systems,man,and cybernetics:systems,2019. [8]ZHANG Weizheng,LI Guoqing,ZHANG Weiwei,et al.A 图3种群规模对所提算法的影响 Fig.3 Impact of population size on the proposed aglroithm cluster based PSO with leader updating mechanism and ring-topology for multimodal multi-objective 4结束语 optimization[J].Swarm and evolutionary computation, 2019.50:100569. 为提高多模态多目标进化算法在搜索过程中 [9]LIANG Jing,XU Weiwei,YUE Caitong,et al.Multimod- 的种群多样性和搜索等价解的能力，本文提出一 al multiobjective optimization with differential 种融合分区和局部搜索的多模态多目标粒子群算 evolution[J].Swarm and evolutionary computation,2019, 法ZLS-SMPSO-MM。在该算法中，首先将整个搜 44:1028-1059 索空间分成多个子空间，然后使用多模态多目标 [10]LIU Yiping,YEN G,GONG Dunwei.A Multimodal mul- 粒子群算法对各个子空间进行搜索，并使用局部 tiobjective evolutionary algorithm using two-archive and 搜索来提高算法的搜索效率，最后将各个子空间 recombination strategies[.IEEE transactions on evolu- 所得解集进行合并选择。为验证ZLS-SMPSO- tionary computation,2019,23(4):660-674. MM算法的性能，选取14个多模态多目标优化问 [11]LIN Qiuzhen,LIN Wu,ZHU Zexuan,et al.Multimodal 题，并将其与DN-NSGA-lⅡ、Omni-optimizer、MO- multiobjective evolutionary optimization with dual clus- Ring-PSO-SCD、SMPSO-MM、ZS-MO-Ring-SCD- tering in decision and objective spaces[J].IEEE transac- tions on evolutionary computation,2021,25(1):130-144. PSO等多模态多目标算法进行比较。实验结果表 [12]ZHANG Xuewei,LIU Hao,TU Liangping.A modified 明，分区搜索和局部搜索能够有效帮助SMPSO: particle swarm optimization for multimodal multi-object- MM找到更多的等价解。 ive optimization[J].Engineering applications of artificial 参考文献： intelligence,2020,95:103905 [13]LI Guosen,YAN Li,QU Boyang.Multi-objective particle [1]BRANKE J.Multiobjective optimization[M].Berlin: swarm optimization based on Gaussian sampling[J].IEEE Springer,2008. access,2020,8:209717-209737 [2]LIANG JJ.YUE C T,QU B Y.Multimodal multi-object- [14]DEB K.Multi-objective evolutionary algorithms:introdu- ive optimization:a preliminary study[C]//Proceedings of cing bias among pareto-optimal solutions[M]//GHOSH A, 2016 IEEE Congress on Evolutionary Computation.Van- TSUTSUI S.Advances in Evolutionary Computing.Ber- couver,Canada:IEEE,2016:2454-2461 lin:Springer,2003. [3]LI Xiaodong.Niching without niching parameters:particle [15]DEB K.Multi-objective genetic algorithms:problem dif- swarm optimization using a ring topology[J].IEEE trans- ficulties and construction of test problems[J].Evolution- actions on evolutionary computation,2010,14(4): ary computation,1999,7(3):205-230. 150-169 [16]ZITZLER E,THIELE L.Multiobjective evolutionary al- [4]YUE Caitong,QU Boyang,LIANG Jing.A multi-object- gorithms:a comparative case study and the strength ive particle swarm optimizer using ring topology for solv- Pareto approach[J].IEEE transactions on evolutionary小，这将直接影响算法的搜索效率。同时，当算 法种群规模在 400 时，算法的性能最佳。但为与 第 3.2 节中比较算法的种群规模保持一致，所提 算法的种群规模仍设置为 800。 3.5 3.0 2.5 2.0 1.5 1.0 排序值 400 800 1 200 1 600 种群规模 2.000 0 2.066 7 2.733 3 3.200 0 图 3 种群规模对所提算法的影响 Fig. 3 Impact of population size on the proposed aglroithm 4 结束语 为提高多模态多目标进化算法在搜索过程中 的种群多样性和搜索等价解的能力，本文提出一 种融合分区和局部搜索的多模态多目标粒子群算 法 ZLS-SMPSO-MM。在该算法中，首先将整个搜 索空间分成多个子空间，然后使用多模态多目标 粒子群算法对各个子空间进行搜索，并使用局部 搜索来提高算法的搜索效率，最后将各个子空间 所得解集进行合并选择。为验证 ZLS-SMPSO￾MM 算法的性能，选取 14 个多模态多目标优化问 题，并将其与 DN-NSGA-II、Omni-optimizer、MO￾Ring-PSO-SCD、SMPSO-MM、ZS-MO-Ring-SCD￾PSO 等多模态多目标算法进行比较。实验结果表 明，分区搜索和局部搜索能够有效帮助 SMPSO￾MM 找到更多的等价解。 参考文献： BRANKE J. Multiobjective optimization[M]. Berlin: Springer, 2008. [1] LIANG J J, YUE C T, QU B Y. Multimodal multi-object￾ive optimization: a preliminary study[C]//Proceedings of 2016 IEEE Congress on Evolutionary Computation. Van￾couver, Canada: IEEE, 2016: 2454−2461. [2] LI Xiaodong. Niching without niching parameters: particle swarm optimization using a ring topology[J]. IEEE trans￾actions on evolutionary computation, 2010, 14(4): 150–169. [3] YUE Caitong, QU Boyang, LIANG Jing. A multi-object￾ive particle swarm optimizer using ring topology for solv- [4] ing multimodal multi-objective problems[J]. IEEE transac￾tions on evolutionary computation, 2018, 22(5): 805–817. LIANG Jing, GUO Qianqian, YUE Caitong, et al. A self￾organizing multi-objective particle swarm optimization al￾gorithm for multimodal multi-objective problems[C]//Pro￾ceedings of the 9th International Conference on Swarm In￾telligence. Shanghai, China: Springer, 2018: 550−560. [5] LI Zhihui, SHI Li, YUE Caitong, et al. Differential evolu￾tion based on reinforcement learning with fitness ranking for solving multimodal multiobjective problems[J]. Swarm and evolutionary computation, 2019, 49: 234–244. [6] FAN Qing, YAN Xuefeng. Solving multimodal multiob￾jective problems through zoning search[J]. IEEE transac￾tions on systems, man, and cybernetics: systems, 2019. [7] ZHANG Weizheng, LI Guoqing, ZHANG Weiwei, et al. A cluster based PSO with leader updating mechanism and ring-topology for multimodal multi-objective optimization[J]. Swarm and evolutionary computation, 2019, 50: 100569. [8] LIANG Jing, XU Weiwei, YUE Caitong, et al. Multimod￾al multiobjective optimization with differential evolution[J]. Swarm and evolutionary computation, 2019, 44: 1028–1059. [9] LIU Yiping, YEN G, GONG Dunwei. A Multimodal mul￾tiobjective evolutionary algorithm using two-archive and recombination strategies[J]. IEEE transactions on evolu￾tionary computation, 2019, 23(4): 660–674. [10] LIN Qiuzhen, LIN Wu, ZHU Zexuan, et al. Multimodal multiobjective evolutionary optimization with dual clus￾tering in decision and objective spaces[J]. IEEE transac￾tions on evolutionary computation, 2021, 25(1): 130–144. [11] ZHANG Xuewei, LIU Hao, TU Liangping. A modified particle swarm optimization for multimodal multi-object￾ive optimization[J]. Engineering applications of artificial intelligence, 2020, 95: 103905. [12] LI Guosen, YAN Li, QU Boyang. Multi-objective particle swarm optimization based on Gaussian sampling[J]. IEEE access, 2020, 8: 209717–209737. [13] DEB K. Multi-objective evolutionary algorithms: introdu￾cing bias among pareto-optimal solutions[M]//GHOSH A, TSUTSUI S. Advances in Evolutionary Computing. Ber￾lin: Springer, 2003. [14] DEB K. Multi-objective genetic algorithms: problem dif￾ficulties and construction of test problems[J]. Evolution￾ary computation, 1999, 7(3): 205–230. [15] ZITZLER E, THIELE L. Multiobjective evolutionary al￾gorithms: a comparative case study and the strength Pareto approach[J]. IEEE transactions on evolutionary [16] 第 4 期 胡洁，等：融合分区和局部搜索的多模态多目标优化 ·783·