第2期 陈强，等：目标空间映射策略的高维多目标粒子群优化算法 ·369· 图4中横坐标为记录的次数，每7次迭代记 swarm optimization with a balanceable fitness estimation 录一次。从中可以看出，在迭代过程中，GD值都 for many-objective optimization problems[J].IEEE trans- 是逐渐减小，算法逐渐收敛。目标映射策略作为 actions on evolutionary computation,2018,22(1):32-46. 一种有效的筛选候选解的方法，对于算法在问题 [8]ZHANG Qingfu,LI Hui.MOEA/D:a multiobjective evol- 中的收敛性能的表现起到决定性的作用，因此， utionary algorithm based on decomposition[J].IEEE trans- 在高维多目标优化问题中，目标映射策略是可行 actions on evolutionary computation,2008,11(6): 712-731. 且有效的。 [9]LIU Hailin,GU Fangqing,ZHANG Qingfu.Decomposi- 4结束语 tion of a multiobjective optimization problem into a num- ber of simple multiobjective subproblems[J].IEEE transac- 本文提出了一种基于目标空间映射策略的高 tions on evolutionary computation,2014,18(3):450-455. 维多目标粒子群算法来求解高维多目标问题。该 [10]LI Ke,DEB K,ZHANG Qingfu,et al.An evolutionary 方法利用性能指标对目标空间进行划分，从而达 many-objective optimization algorithm based on domin- 到增强算法选择压力的目的。通过6组标准测试 ance and decomposition[J.IEEE transactions on evolu- 函数的仿真验证，实验结果表明，在处理高维多 tionary computation,2015,19(5):694-716. 目标问题时，目标空间映射策略能够有效地提高 [11]LIU Songbai.LIN Qiuzhen,TAN K C.et al.A fuzzy de- composition-based Multi/many-objective evolutionary al- 种群的收敛性和分布性。将该算法应用在工程实 例问题将是下一步的研究重点。 gorithm[J].IEEE transactions on cybernetics,2020:1-15. 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[17]LI Miqing,YANG Shengxiang,LIU Xiaohui.Bi-goal [5]汤恺祥，许峰.基于大数据聚类的改进NSGA-Ⅲ算 evolution for many-objective optimization problems[J. 法[).信息记录材料，2020,21(5)：109-112 Artificial intelligence,2015,228:45-65. TANG Kaixiang,XU Feng.Improved NSGA-III al- [18]KENNEDY J,EBERHART R.Particle swarm optimiza- gorithm based on big data clustering[J].Information re- tion[C]//Proceedings of ICNN'95-International Confer- cording materials,2020,21(5):109-112 ence on Neural Network.Perth,Australia,1995: [6]ZOU Juan,FU Liuwei,ZHENG Jinhua,et al.A many-ob- 1942-1948. jective evolutionary algorithm based on rotated grid[J]. [19]SHI Y,EBERHART R C.A modified particle swarm op- Applied soft computing,2018,67:596-609. timizer[C]//1998 IEEE International Conference on Evol- [7]LIN Qiuzhen,LIU Songbai,TANG Chaoyu,et al.Particle utionary Computation Proceedings.IEEE World Con-图 4 中横坐标为记录的次数，每 7 次迭代记 录一次。从中可以看出，在迭代过程中，GD 值都 是逐渐减小，算法逐渐收敛。目标映射策略作为 一种有效的筛选候选解的方法，对于算法在问题 中的收敛性能的表现起到决定性的作用，因此， 在高维多目标优化问题中，目标映射策略是可行 且有效的。 4 结束语 本文提出了一种基于目标空间映射策略的高 维多目标粒子群算法来求解高维多目标问题。该 方法利用性能指标对目标空间进行划分，从而达 到增强算法选择压力的目的。通过 6 组标准测试 函数的仿真验证，实验结果表明，在处理高维多 目标问题时，目标空间映射策略能够有效地提高 种群的收敛性和分布性。将该算法应用在工程实 例问题将是下一步的研究重点。 参考文献： ISHIBUCHI H, TSUKAMOTO N, NOJIMA Y. Evolution￾ary many-objective optimization: a short review[C]//2008 IEEE Congress on Evolutionary Computation (IEEE World Congress on Computational Intelligence). Hong Kong, China, 2008: 266−271. [1] 刘建昌, 李飞, 王洪海, 等. 进化高维多目标优化算法研 究综述 [J]. 控制与决策, 2018, 33(5): 879–887. LIU Jianchang, LI Fei, WANG Honghai, et al. Survey on evolutionary many-objective optimization algorithms[J]. 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Bi-goal evolution for many-objective optimization problems[J]. Artificial intelligence, 2015, 228: 45–65. [17] KENNEDY J, EBERHART R. Particle swarm optimiza￾tion[C]//Proceedings of ICNN'95 - International Confer￾ence on Neural Network. Perth, Australia, 1995: 1942−1948. [18] SHI Y, EBERHART R C. A modified particle swarm op￾timizer[C]//1998 IEEE International Conference on Evol￾utionary Computation Proceedings. IEEE World Con- [19] 第 2 期 陈强，等：目标空间映射策略的高维多目标粒子群优化算法 ·369·