一种采用改进交叉熵的多目标优化问题求解方法
|
摘要
针对传统交叉熵算法不能解决多目标优化问题,采用单目标交叉熵优化算法提出了改进多目标交叉熵优化(Multi-Objective Cross Entropy Optimization,MOCEO)算法。首先,采用个体选择机制来保留进化过程中的优良个体,通过精英保留策略提取优良个体分布信息以不断修正算法正态分布概率模型参数;其次,引入进化方向在正态分布群体采样过程中,引导所产生新个体在解空间中的分布使得种群朝着性能提高的方向进化;最后,为了避免陷入局部最优点在参数平滑操作过程中,定义了调节系数随机调整正态分布概率模型参数。ZDT和DTLZ系列多目标问题的测试结果表明,与经典多目标优化算法NSGA-II、SPEA2、MOEAD、PAES相比,MOCEO在超体积和反转世代距离性能指标以及进化速度等方面较好,是一种收敛速度快、寻优能力强、鲁棒性高的算法。为验证MOCEO在工程实际中的效果,将其应用于某型高速列车悬挂系统横向平稳控制系统的参数优化中,仿真结果表明:相比于NSGA-II算法,使用MOCEO优化调整控制系统参数后,车体横向平稳性指标提高4.16%,横向加速度峰值减小10.34%,横向振动加速度在1～2 Hz人体敏感频率范围内有一定改善,列车具有更好的横向平稳性能。
Due to the traditional cross entropy algorithm is unable to solve multi-objective optimization problem, an improved multi-objective cross entropy optimization(MOCEO) algorithm is proposed based on the single target cross entropy optimization algorithm. The individual selection mechanism is adopted to retain the elite individuals in the evolution process, and the distribution information is extracted via the elite retention strategy to constantly correct the parameters of the normal distribution probability model of the algorithm. Then the evolution direction in the traditional normal distribution population is introduced. During the sampling process, the new individuals are guided to the spatial distribution of the solution so that the population evolves towards the improvement of performance. To avoid the algorithm falling into local optimum, the adjustment coefficient is defined in the process of the traditional parameter smoothing operation. Adjusting the parameters of the normal distribution probability model, premature convergence of the algorithm is effectively avoided. This paper implements the MOCEO and compares it in terms of the hyper volume, the inverted generational distance performance indications and the evolutionary speed with the NSGA-II, SPEA2, MOEAD, PAES algorithms using the ZDT and DTLZ test functions. The results indicate that MOCEO is superior to the other algorithms with fast convergence, strong searching ability and high robustness. An optimization for parameters of horizontal stability control system of a high-speed train suspension system verifies the effect of MOCEO. Compared with the NSGA-II algorithm, the lateral stability index of the train body is increased by 4.16% and the peak value of lateral acceleration is reduced by 10.34% by adjusting the MOCEO optimization parameters of the control system. The lateral vibration acceleration of the vehicle body is improved in the frequency range of 1-2 Hz, to which human body is sensitive, and the train achieves better lateral stability.
Due to the traditional cross entropy algorithm is unable to solve multi-objective optimization problem, an improved multi-objective cross entropy optimization(MOCEO) algorithm is proposed based on the single target cross entropy optimization algorithm. The individual selection mechanism is adopted to retain the elite individuals in the evolution process, and the distribution information is extracted via the elite retention strategy to constantly correct the parameters of the normal distribution probability model of the algorithm. Then the evolution direction in the traditional normal distribution population is introduced. During the sampling process, the new individuals are guided to the spatial distribution of the solution so that the population evolves towards the improvement of performance. To avoid the algorithm falling into local optimum, the adjustment coefficient is defined in the process of the traditional parameter smoothing operation. Adjusting the parameters of the normal distribution probability model, premature convergence of the algorithm is effectively avoided. This paper implements the MOCEO and compares it in terms of the hyper volume, the inverted generational distance performance indications and the evolutionary speed with the NSGA-II, SPEA2, MOEAD, PAES algorithms using the ZDT and DTLZ test functions. The results indicate that MOCEO is superior to the other algorithms with fast convergence, strong searching ability and high robustness. An optimization for parameters of horizontal stability control system of a high-speed train suspension system verifies the effect of MOCEO. Compared with the NSGA-II algorithm, the lateral stability index of the train body is increased by 4.16% and the peak value of lateral acceleration is reduced by 10.34% by adjusting the MOCEO optimization parameters of the control system. The lateral vibration acceleration of the vehicle body is improved in the frequency range of 1-2 Hz, to which human body is sensitive, and the train achieves better lateral stability.
引文
[1] CELESTE F, DAMBREVILLE F, LE CADRE J P. Optimal path planning using cross-entropy method [C]//2006 9th International Conference on Information Fusion. Piscataway, NJ, USA: IEEE, 2006: 1-8.
[2] JALOCON C L C, NERVES A C. Renewable energy portfolio planning using the cross entropy method [C]//Proceedings of 2013 IEEE PES Asia-Pacific Power and Energy Engineering Conference. Piscataway, NJ, USA: IEEE, 2013: 1-5.
[3] CASERTA M, RICO E Q, URIBE A M. A cross entropy algorithm for the Knapsack problem with setups [J]. Computers & Operations Research, 2008, 35(1): 241-252.
[4] HABER R E, DEL TORO R M, GAJATE A. Optimal fuzzy control system using the cross-entropy method: a case study of a drilling process [J]. Information Sciences, 2010, 180(14): 2777-2792.
[5] BOER P T D, KROESE D P, MANNOR S, et al. A tutorial on the cross-entropy method [J]. Annals of Operations Research, 2005, 134(1): 19-67.
[6] AN S, YANG S, HO S L, et al. An improved cross-entropy method applied to inverse problems [J]. IEEE Transactions on Magnetics, 2012, 48(2): 327-330.
[7] LI J, CHAI T Y, GONG J K. Design of PID controller using cross entropy method [J]. Control and Decision, 2011, 26(5): 794-796.
[8] GIAGKIOZIS I, PURSHOUSE R C, FLEMING P J. Generalized decomposition and cross entropy methods for many-objective optimization [J]. Information Sciences, 2014, 282: 363-387.
[9] 边莉, 车向前, 张少卿. 基于改进交叉熵算法多目标不等间距阵列综合 [J]. 上海交通大学学报, 2014, 48(3): 372-376. BIAN Li, CHE Xiangqian, ZHANG Shaoqing. Multi-objective unequally-spaced array synthesis based on modified cross entropy algorithm [J]. Journal of Shanghai Jiaotong University, 2014, 48(3): 372-376.
[10] BERUVIDES G, QUIZA R, HABER R E. Multi-objective optimization based on an improved cross-entropy method: a case study of a micro-scale manufacturing process [J]. Information Sciences, 2016, 334: 161-173.
[11] 陈春俊. 高速列车横向主动/半主动悬挂控制研究 [D]. 成都: 西南交通大学, 2006: 49-65.
[12] 孟宏. 提速机车车辆横向运动稳定性研究及应用 [D]. 成都: 西南交通大学, 2009: 10-20.
[13] 周洪涛, 杨绍普, 朱红霞. 基于模糊控制的高速机车横向半主动控制研究 [J]. 振动与冲击, 2011, 30(9): 145-149. ZHOU Hongtao, YANG Shaopu, ZHU Hongxia. Fuzzy control strategy applied in lateral semi-active control of the high-speed locomotive [J]. Journal of Vibration and Shock, 2011, 30(9): 145-149.
[14] DEB K, PRATAP A, AGAREAL S, et al. A fast and elitist multiobjective genetic algorithm: NSGA-II [J]. IEEE Transactions on Evolutionary Computation, 2002, 6(2): 182-197.
[15] ZIZTLER E, LAUMANNS M, THIELE L. SPEA2: improving the strength Pareto evolutionary algorithm for multi-objective optimization: TIK-Report 103 [R]. Zurich, Switzerland: Swiss Federal Institute of Technology, 2001: 1-21.
[16] ZHANG Q, LI H. MOEA/D: a multi objective evolutionary algorithm based on decomposition [J]. IEEE Transactions on evolutionary computation, 2007, 11(6): 712-731.
[17] KNOWLES J D, CORNE D W. Approximating the nondominated front using the Pareto archived evolution strategy [J]. Evolutionary Computation, 2000, 8(2): 149-172.
[18] RUBINSTEIN R Y. Optimization of computer simulation models with rare events [J]. European Journal of Operational Research, 1997, 99(1): 89-112.
[19] KROESE D P, POROTSKY S, RUBINSTEIN R Y. The cross-entropy method for continuous multi-extremal optimization [J]. Methodology and Computing in Applied Probability, 2006, 8(3): 383-407.
[20] 邓涛, 林椿松, 李亚南, 等. 采用NSGA-II算法的混合动力能量管理控制多目标优化方法 [J]. 西安交通大学学报, 2015, 49(10): 143-150. DENG Tao, LIN Chunsong, LI Yanan, et al. A multi-objective optimization method for energy management control of hybrid electric vehicles using NSGA-II algorithm [J]. Journal of Xi’an Jiaotong University, 2015, 49(10): 143-150.
[21] ZITZLER E, DEB K, THIELE L. Comparison of multi-objective evolutionary algorithms: empirical results [J]. Evolutionary Computation, 2000, 8(2): 173-195.
[22] RUSSO L M S, FRANCISCO A P. Quick hypervolume [J]. IEEE Transactions on Evolutionary Computation, 2014, 18(4): 481-502.
[23] DURILLO J J, NEBRO A J. jMetal: a Java framework for multi-objective optimization [J]. Advances in Engineering Software, 2011, 42(10): 760-771.
[24] 李中奇, 杨振村, 杨辉, 等. 高速列车双自适应广义预测控制方法 [J]. 中国铁道科学, 2015, 36(6): 120-127. LI Zhongqi, YANG Zhencun, YANG Hui. Generalized predictive control with dual adaptation method of high speed train [J]. China Railway Science, 2015, 36(6): 120-127.
[25] 杨世杰, 陈建政. Matlab在机车车辆动力学试验数据处理中的应用 [J]. 中国测试技术, 2006, 32(3): 26-28. YANG Shijie, CHEN Jianzheng. Data processing in the dynamic test of rail vehicle based on Matlab [J]. China Measurement & Test, 2006, 32(3): 26-28.
[2] JALOCON C L C, NERVES A C. Renewable energy portfolio planning using the cross entropy method [C]//Proceedings of 2013 IEEE PES Asia-Pacific Power and Energy Engineering Conference. Piscataway, NJ, USA: IEEE, 2013: 1-5.
[3] CASERTA M, RICO E Q, URIBE A M. A cross entropy algorithm for the Knapsack problem with setups [J]. Computers & Operations Research, 2008, 35(1): 241-252.
[4] HABER R E, DEL TORO R M, GAJATE A. Optimal fuzzy control system using the cross-entropy method: a case study of a drilling process [J]. Information Sciences, 2010, 180(14): 2777-2792.
[5] BOER P T D, KROESE D P, MANNOR S, et al. A tutorial on the cross-entropy method [J]. Annals of Operations Research, 2005, 134(1): 19-67.
[6] AN S, YANG S, HO S L, et al. An improved cross-entropy method applied to inverse problems [J]. IEEE Transactions on Magnetics, 2012, 48(2): 327-330.
[7] LI J, CHAI T Y, GONG J K. Design of PID controller using cross entropy method [J]. Control and Decision, 2011, 26(5): 794-796.
[8] GIAGKIOZIS I, PURSHOUSE R C, FLEMING P J. Generalized decomposition and cross entropy methods for many-objective optimization [J]. Information Sciences, 2014, 282: 363-387.
[9] 边莉, 车向前, 张少卿. 基于改进交叉熵算法多目标不等间距阵列综合 [J]. 上海交通大学学报, 2014, 48(3): 372-376. BIAN Li, CHE Xiangqian, ZHANG Shaoqing. Multi-objective unequally-spaced array synthesis based on modified cross entropy algorithm [J]. Journal of Shanghai Jiaotong University, 2014, 48(3): 372-376.
[10] BERUVIDES G, QUIZA R, HABER R E. Multi-objective optimization based on an improved cross-entropy method: a case study of a micro-scale manufacturing process [J]. Information Sciences, 2016, 334: 161-173.
[11] 陈春俊. 高速列车横向主动/半主动悬挂控制研究 [D]. 成都: 西南交通大学, 2006: 49-65.
[12] 孟宏. 提速机车车辆横向运动稳定性研究及应用 [D]. 成都: 西南交通大学, 2009: 10-20.
[13] 周洪涛, 杨绍普, 朱红霞. 基于模糊控制的高速机车横向半主动控制研究 [J]. 振动与冲击, 2011, 30(9): 145-149. ZHOU Hongtao, YANG Shaopu, ZHU Hongxia. Fuzzy control strategy applied in lateral semi-active control of the high-speed locomotive [J]. Journal of Vibration and Shock, 2011, 30(9): 145-149.
[14] DEB K, PRATAP A, AGAREAL S, et al. A fast and elitist multiobjective genetic algorithm: NSGA-II [J]. IEEE Transactions on Evolutionary Computation, 2002, 6(2): 182-197.
[15] ZIZTLER E, LAUMANNS M, THIELE L. SPEA2: improving the strength Pareto evolutionary algorithm for multi-objective optimization: TIK-Report 103 [R]. Zurich, Switzerland: Swiss Federal Institute of Technology, 2001: 1-21.
[16] ZHANG Q, LI H. MOEA/D: a multi objective evolutionary algorithm based on decomposition [J]. IEEE Transactions on evolutionary computation, 2007, 11(6): 712-731.
[17] KNOWLES J D, CORNE D W. Approximating the nondominated front using the Pareto archived evolution strategy [J]. Evolutionary Computation, 2000, 8(2): 149-172.
[18] RUBINSTEIN R Y. Optimization of computer simulation models with rare events [J]. European Journal of Operational Research, 1997, 99(1): 89-112.
[19] KROESE D P, POROTSKY S, RUBINSTEIN R Y. The cross-entropy method for continuous multi-extremal optimization [J]. Methodology and Computing in Applied Probability, 2006, 8(3): 383-407.
[20] 邓涛, 林椿松, 李亚南, 等. 采用NSGA-II算法的混合动力能量管理控制多目标优化方法 [J]. 西安交通大学学报, 2015, 49(10): 143-150. DENG Tao, LIN Chunsong, LI Yanan, et al. A multi-objective optimization method for energy management control of hybrid electric vehicles using NSGA-II algorithm [J]. Journal of Xi’an Jiaotong University, 2015, 49(10): 143-150.
[21] ZITZLER E, DEB K, THIELE L. Comparison of multi-objective evolutionary algorithms: empirical results [J]. Evolutionary Computation, 2000, 8(2): 173-195.
[22] RUSSO L M S, FRANCISCO A P. Quick hypervolume [J]. IEEE Transactions on Evolutionary Computation, 2014, 18(4): 481-502.
[23] DURILLO J J, NEBRO A J. jMetal: a Java framework for multi-objective optimization [J]. Advances in Engineering Software, 2011, 42(10): 760-771.
[24] 李中奇, 杨振村, 杨辉, 等. 高速列车双自适应广义预测控制方法 [J]. 中国铁道科学, 2015, 36(6): 120-127. LI Zhongqi, YANG Zhencun, YANG Hui. Generalized predictive control with dual adaptation method of high speed train [J]. China Railway Science, 2015, 36(6): 120-127.
[25] 杨世杰, 陈建政. Matlab在机车车辆动力学试验数据处理中的应用 [J]. 中国测试技术, 2006, 32(3): 26-28. YANG Shijie, CHEN Jianzheng. Data processing in the dynamic test of rail vehicle based on Matlab [J]. China Measurement & Test, 2006, 32(3): 26-28.