无数学模型的非线性约束单目标系统优化方法改进
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摘要
为提高无法准确建立数学模型的非线性约束单目标系统优化问题的寻优精度,并考虑获取样本的代价,提出一种基于支持向量机和免疫粒子群算法的组合方法(support vector machine and immune particle swarm optimization,SVM-IPSO).首先,运用支持向量机构建非线性约束单目标系统预测模型,然后,采用引入了免疫系统自我调节机制的免疫粒子群算法在预测模型的基础上对系统寻优.与基于BP神经网络和粒子群算法的组合方法(BP and particle swarm optimization,BP-PSO)进行仿真实验对比,同时,通过减少训练样本,研究了在训练样本较少情况下两种方法的寻优效果.实验结果表明,在相同样本数量条件下,SVM-IPSO方法具有更高的优化能力,并且当样本数量减少时,相比BP-PSO方法,SVM-IPSO方法仍能获得更稳定且更准确的系统寻优值.因此,SVM-IPSO方法为实际中此类问题提供了一个新的更优的解决途径.
Optimization problems of nonlinear constrained single objective system are common in engineering and many other fields. Considering practical applications,many optimization methods have been proposed to optimize such systems whose accurate mathematical models are easily constructed. However,as more variables are being considered in practical applications,objective systems are becoming more complex,so that corresponding accurate mathematical models are difficult to be constructed. Many previous scholars mainly used back propagation( BP) neural network and basic optimization algorithms to successfully solve systems that are without accurate mathematical models. But the optimization accuracy still needs to be further improved. In addition,samples are needed to solve such system optimization problems. Therefore,to improve the optimization accuracy of nonlinear constrained single objective systems that are without accurate mathematical models while considering the cost of obtaining samples,a new method based on a combination of support vector machine and immune particle swarm optimization algorithm( SVM-IPSO) is proposed. First,the SVM is used to construct the predicted model of nonlinear constrained single objective system. Then,the immune particle swarm algorithm,which incorporates the self-regulatory mechanism of the immune system,is used to optimize the system based on the predicted model.The proposed method is compared with a method based on a combination of BP neural network and particle swarm optimization algorithm( BP-PSO). The optimization effects of the two methods are studied under few training samples by reducing the number of training sam-ples. The simulation results show that the SVM-IPSO has a higher optimization ability under the same sample size conditions,and when the number of samples decreases,the SVM-IPSO method can still obtain more stable and accurate system optimization values than the BP-PSO method. Hence,the SVM-IPSO method provides a new and better solution to this kind of problems.
Optimization problems of nonlinear constrained single objective system are common in engineering and many other fields. Considering practical applications,many optimization methods have been proposed to optimize such systems whose accurate mathematical models are easily constructed. However,as more variables are being considered in practical applications,objective systems are becoming more complex,so that corresponding accurate mathematical models are difficult to be constructed. Many previous scholars mainly used back propagation( BP) neural network and basic optimization algorithms to successfully solve systems that are without accurate mathematical models. But the optimization accuracy still needs to be further improved. In addition,samples are needed to solve such system optimization problems. Therefore,to improve the optimization accuracy of nonlinear constrained single objective systems that are without accurate mathematical models while considering the cost of obtaining samples,a new method based on a combination of support vector machine and immune particle swarm optimization algorithm( SVM-IPSO) is proposed. First,the SVM is used to construct the predicted model of nonlinear constrained single objective system. Then,the immune particle swarm algorithm,which incorporates the self-regulatory mechanism of the immune system,is used to optimize the system based on the predicted model.The proposed method is compared with a method based on a combination of BP neural network and particle swarm optimization algorithm( BP-PSO). The optimization effects of the two methods are studied under few training samples by reducing the number of training sam-ples. The simulation results show that the SVM-IPSO has a higher optimization ability under the same sample size conditions,and when the number of samples decreases,the SVM-IPSO method can still obtain more stable and accurate system optimization values than the BP-PSO method. Hence,the SVM-IPSO method provides a new and better solution to this kind of problems.
引文
[1] Courant R. Variational methods for the solution of problems of equilibrium and vibrations. Bull Am Math Soc,1943,49(1943):1
[2] Powell M J D. A method for nonlinear constraints in minimization problems. Optimization,1969,5(6):283
[3] Yang J Z,Sun Q Y,Wang J,et al. Robust principal component analysis based on advanced augmented lagrange multiplier method.J Harbin Inst Technol,2015,47(11):27(杨剑哲,孙巧榆,王君,等.基于改进增广拉格朗日乘子法的鲁棒性主成分分析.哈尔滨工业大学学报,2015,47(11):27)
[4] Mc Dougall T J, Wotherspoon S J. A simple modification of Newton's method to achieve convergence of ord槡er 1+2. Appl Math Lett,2014,29:20
[5] Shen C G,Zhang L H,Liu W. A stabilized filter SQP algorithm for nonlinear programming. J Global Optimization,2016,65(4):677
[6] Ruan M Z,Li Q M,Wang H J,et al. Application of artificial immune particle-swarm-optimization algorithm to system-reliability optimization. Control Theory Appl,2010,27(9):1253(阮旻智,李庆民,王红军,等.人工免疫粒子群算法在系统可靠性优化中的应用.控制理论与应用,2010,27(9):1253)
[7] Cheng Y,Cheng W M,Zheng Y,et al. Structural reliability optimal design based on chaos particle swarm optimization. J Central South Univ Sci Technol,2011,42(3):671(程跃,程文明,郑严,等.基于混沌粒子群算法的结构可靠性优化设计.中南大学学报(自然科学版),2011,42(3):671)
[8] Beheshti Z,Shamsuddin S M,Yuhaniz S S. Binary accelerated particle swarm algorithm(BAPSA)for discrete optimization problems. J Global Optimization,2013,57(2):549
[9] Chang H W,Ma H,Zhang J Q,et al. Optimization of metamaterial based weighted real-coded genetic algorithm. Acta Phys Sin,2014,63(8):087804-1(常红伟,马华,张介秋,等.基于加权实数编码遗传算法的超材料优化设计.物理学报,2014,63(8):087804-1)
[10] Xu M X,Zhang X S,Yu T. Transfer bees optimizer and its application on reactive power optimization. Acta Autom Sinica,2017,43(1):83(徐茂鑫,张孝顺,余涛.迁移蜂群优化算法及其在无功优化中的应用.自动化学报,2017,43(1):83)
[11] Qiang P. The Research of Perfect Matching of Tunneling Parameter for Super Large Diameter Earth Pressure Balance Machine[Dissertation]. Shanghai:Shanghai University,2014(羌培.超大直径土压平衡盾构最佳施工参数匹配研究[学位论文].上海:上海大学,2014)
[12] Liu L L,Zhang Q H,Li C Y,et al. Optimization research on rough rolling process of H-beam based on ANN and GA. Forging Stamping Technol,2011,36(1):153(刘兰兰,张勤河,李传宇,等.基于神经网络和遗传算法的H型钢粗轧工艺优化.锻压技术,2011,36(1):153)
[13] Ni L B,Liu J C,Wu Y T,et al. Optimization of laser cladding process variables based on neural network and particle swarm optimization algorithms. Chin J Lasers,2011,38(2):0203003-1(倪立斌,刘继常,伍耀庭,等.基于神经网络和粒子群算法的激光熔覆工艺优化.中国激光,2011,38(2):0203003-1)
[14] Wang Q P. The Density Control of the MIM Injection Molded Billet Based on the Particle Swarm Optimization and Neural Networks[Dissertation]. Changsha:Central South University,2014(王秋平.基于粒子群算法和神经网络的MIM坯体密度分布控制[学位论文].长沙:中南大学,2014)
[15] Yan B Y,Sheng Z F,Li G,et al. Inverse method based on neural network and particle swarm optimization for characterizing the C/C material elastic constants. J Solid Rocket Technol,2016,39(1):106(严博燕,生志斐,李耿,等.一种基于神经网络与粒子群算法辨识针刺炭/炭复合材料弹性常数的方法.固体火箭技术,2016,39(1):106)
[16] Zhang H G,Zhang S,Yin Y X. Multi-class fault diagnosis of BF based on global optimization LS-SVM. Chin J Eng,2017,39(1):39(张海刚,张森,尹怡欣.基于全局优化支持向量机的多类别高炉故障诊断.工程科学学报,2017,39(1):39)
[17] Wu J L,Yang S X,Liu C S. Parameter selection for support vector machines based on genetic algorithms to short-term power load forecasting. J Central South Univ Sci Technol,2009,40(1):180(吴景龙,杨淑霞,刘承水.基于遗传算法优化参数的支持向量机短期负荷预测方法.中南大学学报(自然科学版),2009,40(1):180)
[18] Eberhart R,Kennedy J. A new optimizer using particle swarm theory//Proceedings of the Sixth International Symposium on Micro Machine and Human Science. Nagoya,1995:39
[19] Kathiravan R,Ganguli R. Strength design of composite beam using gradient and particle swarm optimization. Compos Struct,2007,81(4):471
[20] Zhang C K,Li X F,Shao H H. Chaos optimization algorithm based on linear search and its application to nonlinear constraint optimization problems. Control Decision,2001,16(1):123(张春慨,李霄峰,邵惠鹤.基于线性搜索的混沌优化及其在非线性约束优化问题中的应用.控制与决策,2001,16(1):123)
[21] Wang X C,Shi F,Yu L,et al. 43 Cases Analysis of MATLAB Neural Network. Beijing:Beihang University Press,2013(王小川,史峰,郁磊,等. MATLAB神经网络43个案例分析.北京:北京航空航天大学出版社,2013)
[2] Powell M J D. A method for nonlinear constraints in minimization problems. Optimization,1969,5(6):283
[3] Yang J Z,Sun Q Y,Wang J,et al. Robust principal component analysis based on advanced augmented lagrange multiplier method.J Harbin Inst Technol,2015,47(11):27(杨剑哲,孙巧榆,王君,等.基于改进增广拉格朗日乘子法的鲁棒性主成分分析.哈尔滨工业大学学报,2015,47(11):27)
[4] Mc Dougall T J, Wotherspoon S J. A simple modification of Newton's method to achieve convergence of ord槡er 1+2. Appl Math Lett,2014,29:20
[5] Shen C G,Zhang L H,Liu W. A stabilized filter SQP algorithm for nonlinear programming. J Global Optimization,2016,65(4):677
[6] Ruan M Z,Li Q M,Wang H J,et al. Application of artificial immune particle-swarm-optimization algorithm to system-reliability optimization. Control Theory Appl,2010,27(9):1253(阮旻智,李庆民,王红军,等.人工免疫粒子群算法在系统可靠性优化中的应用.控制理论与应用,2010,27(9):1253)
[7] Cheng Y,Cheng W M,Zheng Y,et al. Structural reliability optimal design based on chaos particle swarm optimization. J Central South Univ Sci Technol,2011,42(3):671(程跃,程文明,郑严,等.基于混沌粒子群算法的结构可靠性优化设计.中南大学学报(自然科学版),2011,42(3):671)
[8] Beheshti Z,Shamsuddin S M,Yuhaniz S S. Binary accelerated particle swarm algorithm(BAPSA)for discrete optimization problems. J Global Optimization,2013,57(2):549
[9] Chang H W,Ma H,Zhang J Q,et al. Optimization of metamaterial based weighted real-coded genetic algorithm. Acta Phys Sin,2014,63(8):087804-1(常红伟,马华,张介秋,等.基于加权实数编码遗传算法的超材料优化设计.物理学报,2014,63(8):087804-1)
[10] Xu M X,Zhang X S,Yu T. Transfer bees optimizer and its application on reactive power optimization. Acta Autom Sinica,2017,43(1):83(徐茂鑫,张孝顺,余涛.迁移蜂群优化算法及其在无功优化中的应用.自动化学报,2017,43(1):83)
[11] Qiang P. The Research of Perfect Matching of Tunneling Parameter for Super Large Diameter Earth Pressure Balance Machine[Dissertation]. Shanghai:Shanghai University,2014(羌培.超大直径土压平衡盾构最佳施工参数匹配研究[学位论文].上海:上海大学,2014)
[12] Liu L L,Zhang Q H,Li C Y,et al. Optimization research on rough rolling process of H-beam based on ANN and GA. Forging Stamping Technol,2011,36(1):153(刘兰兰,张勤河,李传宇,等.基于神经网络和遗传算法的H型钢粗轧工艺优化.锻压技术,2011,36(1):153)
[13] Ni L B,Liu J C,Wu Y T,et al. Optimization of laser cladding process variables based on neural network and particle swarm optimization algorithms. Chin J Lasers,2011,38(2):0203003-1(倪立斌,刘继常,伍耀庭,等.基于神经网络和粒子群算法的激光熔覆工艺优化.中国激光,2011,38(2):0203003-1)
[14] Wang Q P. The Density Control of the MIM Injection Molded Billet Based on the Particle Swarm Optimization and Neural Networks[Dissertation]. Changsha:Central South University,2014(王秋平.基于粒子群算法和神经网络的MIM坯体密度分布控制[学位论文].长沙:中南大学,2014)
[15] Yan B Y,Sheng Z F,Li G,et al. Inverse method based on neural network and particle swarm optimization for characterizing the C/C material elastic constants. J Solid Rocket Technol,2016,39(1):106(严博燕,生志斐,李耿,等.一种基于神经网络与粒子群算法辨识针刺炭/炭复合材料弹性常数的方法.固体火箭技术,2016,39(1):106)
[16] Zhang H G,Zhang S,Yin Y X. Multi-class fault diagnosis of BF based on global optimization LS-SVM. Chin J Eng,2017,39(1):39(张海刚,张森,尹怡欣.基于全局优化支持向量机的多类别高炉故障诊断.工程科学学报,2017,39(1):39)
[17] Wu J L,Yang S X,Liu C S. Parameter selection for support vector machines based on genetic algorithms to short-term power load forecasting. J Central South Univ Sci Technol,2009,40(1):180(吴景龙,杨淑霞,刘承水.基于遗传算法优化参数的支持向量机短期负荷预测方法.中南大学学报(自然科学版),2009,40(1):180)
[18] Eberhart R,Kennedy J. A new optimizer using particle swarm theory//Proceedings of the Sixth International Symposium on Micro Machine and Human Science. Nagoya,1995:39
[19] Kathiravan R,Ganguli R. Strength design of composite beam using gradient and particle swarm optimization. Compos Struct,2007,81(4):471
[20] Zhang C K,Li X F,Shao H H. Chaos optimization algorithm based on linear search and its application to nonlinear constraint optimization problems. Control Decision,2001,16(1):123(张春慨,李霄峰,邵惠鹤.基于线性搜索的混沌优化及其在非线性约束优化问题中的应用.控制与决策,2001,16(1):123)
[21] Wang X C,Shi F,Yu L,et al. 43 Cases Analysis of MATLAB Neural Network. Beijing:Beihang University Press,2013(王小川,史峰,郁磊,等. MATLAB神经网络43个案例分析.北京:北京航空航天大学出版社,2013)