工程科学学报，第40卷，第11期：1402-1411,2018年11月 Chinese Journal of Engineering,Vol.40,No.11:1402-1411,November 2018 DOI:10.13374/j.issn2095-9389.2018.11.014;http://journals.ustb.edu.cn 无数学模型的非线性约束单目标系统优化方法改进 侯公羽2)，许哲东1)区，刘欣)，牛晓同)，王清乐) 1)中国矿业大学（北京）力学与建筑工程学院，北京1000832)新疆工程学院矿业工程与地质学院，乌鲁木齐830091 ☒通信作者，E-mail:18310676138@163.com 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-PSO).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-IPS0方法具有更高的优化能力，并且当样本数量减少时，相比BP-PS0方法，SVM-PS0方法仍能获得更稳定 且更准确的系统寻优值.因此，SVM-PS0方法为实际中此类问题提供了一个新的更优的解决途径. 关键词非线性约束单目标系统：支持向量机；免疫粒子群算法；仿真：优化 分类号TP301.6 Optimization method improvement for nonlinear constrained single objective system without mathematical models HOU Gong-yu),XU Zhe-dong,LIU Xin,NIU Xiao-tong,WANG Qing-le 1)School of Mechanics and Civil Engineering,China University of Mining &Technology (Beijing),Beijing 100083,China 2)School of Mining Engineering and Geology,Xinjiang Institute of Engineering.Urumqi 830091,China Corresponding author,E-mail:18310676138@163.com ABSTRACT 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 sys- tems 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 nee- ded to solve such system optimization problems.Therefore,to improve the optimization accuracy of nonlinear constrained single objec- tive 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- 收稿日期：2017-10-20 基金项目：国家自然科学基金委员会与神华集团有限责任公司联合重点资助项目(U1261212,U1361210):国家自然科学基金面上资助项目 (51574247)工程科学学报,第 40 卷,第 11 期:1402鄄鄄1411,2018 年 11 月 Chinese Journal of Engineering, Vol. 40, No. 11: 1402鄄鄄1411, November 2018 DOI: 10. 13374 / j. issn2095鄄鄄9389. 2018. 11. 014; http: / / journals. ustb. edu. cn 无数学模型的非线性约束单目标系统优化方法改进 侯公羽1,2) , 许哲东1)苣 , 刘 欣1) , 牛晓同1) , 王清乐1) 1) 中国矿业大学(北京)力学与建筑工程学院, 北京 100083 2) 新疆工程学院矿业工程与地质学院, 乌鲁木齐 830091 苣 通信作者, E鄄mail: 18310676138@ 163. com 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鄄鄄 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鄄鄄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鄄鄄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鄄鄄PSO 方法,SVM鄄鄄IPSO 方法仍能获得更稳定 且更准确的系统寻优值. 因此,SVM鄄鄄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. 关键词 非线性约束单目标系统; 支持向量机; 免疫粒子群算法; 仿真; 优化 分类号 TP301郾 6 收稿日期: 2017鄄鄄10鄄鄄20 基金项目: 国家自然科学基金委员会与神华集团有限责任公司联合重点资助项目(U1261212,U1361210); 国家自然科学基金面上资助项目 (51574247) Optimization method improvement for nonlinear constrained single objective system without mathematical models HOU Gong鄄yu 1,2) , XU Zhe鄄dong 1)苣 , LIU Xin 1) , NIU Xiao鄄tong 1) , WANG Qing鄄le 1) 1) School of Mechanics and Civil Engineering, China University of Mining & Technology (Beijing), Beijing 100083, China 2) School of Mining Engineering and Geology, Xinjiang Institute of Engineering, Urumqi 830091, China 苣 Corresponding author, E鄄mail: 18310676138@ 163. com ABSTRACT 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 sys鄄 tems 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 nee鄄 ded to solve such system optimization problems. Therefore, to improve the optimization accuracy of nonlinear constrained single objec鄄 tive 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鄄鄄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鄄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鄄鄄PSO). The optimization effects of the two methods are studied under few training samples by reducing the number of training sam鄄