基于元模型的全局优化算法研究
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摘要
现代机电产品日趋复杂，对其进行仿真分析需要耗费越来越多的计算资源。尽管计算机技术日新月异，计算速度越来越快，但仍不能满足工业界对仿真分析的需求。有报道称，对汽车碰撞模型进行一次仿真分大致需要36－160小时，要实现该模型的两个变量的设计优化则需要75天到11个月，这在实际工程应用中是不可接受的。为应对巨大的挑战，在过去的20年中，元模型方法应运而生并在工业界得到了普遍应用，该方法能够在不影响仿真目标模型精度的情况下减少优化迭代的仿真次数，从而减少对计算资源的消耗。
元模型方法是指利用实验设计所产生的采样点构造近似简化模型代替复杂昂贵的仿真目标模型进行优化分析的优化方法，基于元模型的全局优化则是在元模型方法基础上利用全局优化算法搜索最优点，涉及到实验设计方法、元模型方法、全局优化算法等方面的研究。本文围绕自适应序列采样方法、DIRECT全局优化算法、增量元模型方法以及Pareto多目标优化方法开展了进一步的研究，具体包括以下几个方面：
1）基于RBF元模型的自适应序列采样方法。由于很难确定合适的采样点数量，一次采样构造一个元模型通常是不合理的。自适应序列采样通过逐次增加采样并使其尽量分布于最合理位置，从而克服了一次采样的缺陷，达到以少量点构造精确元模型的效果。本文利用RBF元模型表达式比较简单，易于计算采样点处的曲率的特点，提出了以采样点位置最大曲率和采样点之间最小距离的作为采样标准。新采样方法能使新增采样点分布于影响元模型精度的波峰与波谷处，达到用较少的点反映了复杂目标模型几何特征的效果。以5个常用测试函数以及弹簧和焊接梁的最优设计为例测试新采样方法效果，并与拉丁超立方采样与网格采样进行比较。实验结果表明了该方法的精确性和有效性，而且优于其他采样方法。
2）基于元模型的DIRECT全局优化算法。本文在深入分析标准DIRECT算法原理和收敛性的基础上，针对标准DIRECT算法函数估值次数多、收敛速度慢的缺点，提出基于元模型的改进DIRECT算法。该方法利用优化过程中每次迭代所产生采样点来构造近似元模型，并在其上搜索最优点，从而加快了算法的收敛速度。将改进后的DIRECT算法应用于压力容器的优化设计，并对5种常用元模型的加速效果进行比较。实验结果表明RBF元模型对提高DIRECT算法的收敛性效果最好。
3）基于增量RBF元模型方法的全局优化算法。研究了增量拉丁超立方采样方法与增量径向基函数元模型更新方法；针对基于减法和加法的增量拉丁超立方采样方法难以控制采样点数量以及新增点必须是原有采样点整数倍的缺陷，改进了增量拉丁超立方采样方法；在此基础上，结合增量RBF元模型方法提出一种全新的全局优化方法。以2种常用测试函数与焊接梁为例进行测试，实验结果证明了该方法的可行性和有效性。
4）基于元模型的多目标优化方法。文本针对现有基于元模型的多目标方法大多是单值元模型的缺点提出响应面集的概念。响应面集利用RBF元模型表达式为线性方程的特点，将原有的系数向量转换为系数矩阵，使得多个子目标函数可与同一个元模型相对应。针对大规模采样直接计算Pareto适应度困难的问题，本文提出一种可增量更新的迭代式Pareto适应度计算方法。该方法克服了采样点过多导致Pareto适应度矩阵庞大的问题，并且充分利用上一次迭代中所产生的适应度值信息进行增量更新，从而减少了计算量。将响应面集与增量Pareto适应度计算方法集成到基于元模型的多目标优化算法中，并应用于两杆对称桁架与I字衡量的优化设计，实验结果表明了该方法的工程实用性与有效性。
最后，对本文的主要研究成果和创新点进行了回顾和总结，并对基于元模型的全局优化下一步研究热点和未来发展方向进行了探讨。
Mordern mechanical and electrical products become more and more complex, and it need more and more computing resource to simulate and analysis these products. Although computer technology developed rapidly, and the speed of computer become faster, it hard to meet the industry demand for simulation and analysis. It reported that an analysis of simulating automobile collision need36-160hours to simulate, and an optimization of2variables will cost almost from75days to11months. It is hard to acceptable in practical engineering applications. To deal with the challenges, the metamodel method was widely used in industry product design. This method could cut the simulation times of simulation without affecting the accuracy of simuation target model, and reduce the consumption of computing resources.
The method of metamodel is that replace complex and expensive simulation target model with simplified model which constructed by sampling points generated through experiment design.The global optimization based on metamodel is that use metamodel to search global optimum and it include the research of design of experiment, metamodel, global optimization. This article will discuss the adaptively sequential sampling, DIRECT algorithm, and incremental metamodel and Pareto multi-objective optimization and it include this content as follows.
1) Adaptively sequential sampling based on RBF metamodel. Because of hard to determin the number of sampling points, it is commonly unreasonable that construct metamode through one time sampling. To construct metamodel with a few points and overcome the defect of one time sampling method, adaptively sequential sampling increased points and make it at reasonable position as fas as possible. In order to utilize the geometrical feature of metamodel, maximum curvature of the response surface and minimum distance among the sampling sites, as a general sampling criterion, are adopted in sequential sampling procedure. For the simplicity of RBF model, we can easily evaluate curvature on design optimum via computing the difference and Hessian matrix. A new model approximation algorithm integrated sequential optimal sampling is presented. To illustrate the efficiency and accuracy of the proposed algorithm,5benchmark function and stress spring and weld beam problem have been tested, the result show that the new method is better than latin hyper-cube design and grid sampling.
2) DIRECT algorithm based on metamodel. Based on deeply analysing the principle and convergence properties of DIRECT algorithm, to overcome the defect of DIRECT method, which is too much evaluation of objective function and slow convergence rate, a modified DIRECT method based on radial basis functions was proposed. The presented method identifies the optimum area containing the global or local optimum through analysing the information of sampling points. It enhances the convergence rate of DIRECT method through constructing the radial basis functions on the sampling points which collected from optimum area and searching global optimum on the metamodel. The presented method is applied to some numerical examples and a pressure vessel design and compare accelerated effects with5common metamodel.The results of these examples demonstrate that the best metamodel is RBF metamodel.
3) A new global optimization method based on incremental metamodel. Based on research of incremental Latin Hyper-cube design and incremental updating of RBF metamodel. A new global optimization algorithm was proposed for the complex simulation model problem. To overcome the defect of old incremental latin hyper-cube sampling, which is hard to control the number of sampling points and limited to multiples, we proposed a improved incremental latin hyper-cube sampling method based on subtraction rule ideal. Combined incremental Latin hyper-cube sampling and the method of incremental update RBF metamodel, we proposed a new efficient global optimization algorithm. The presented method was applied in2commonly test functions and a weld beam problem. The results of the example demonstrated the efficiency and engineering practicability of the presented method.
4) Multi-objective optimization based on metamodel. To overcome the defect of the old multi-objective opitimization based on metamodel which used single value metamodel, a subgoal function must construct a correspond metatmodel. We proposed the conception of response surface set; which utilize the linear feature of RBF metamodel and converte the coefficient vector to coefficient matrix, made multiple subgoals function mapping one metamodel. It was hard to directly calculate Pareto fitness of massive sampling points; In order to make full use of information of sampling points generated by last iteration and reduce the computational complexity of optimization algorithm, we proposed a new incremental updating method for calculate Pareto fitness, which overcomes the defect of large-scale matrix caused by excessive sampling points. A new multi-objective optimization algorithm based on metamodel integrated response surface set and method of incrementally calculating Pareto fitness is proposed. The presented method was applied two-bar truss design problem and I-beam design. The results of the example demonstrated the efficiency and engineering practicability of the presented method.
Finally, the main research findings and innovations have been viewed and summarized at the end the article, and the challenges and future development of global optimization based on metamodel have been discussed.
元模型方法是指利用实验设计所产生的采样点构造近似简化模型代替复杂昂贵的仿真目标模型进行优化分析的优化方法，基于元模型的全局优化则是在元模型方法基础上利用全局优化算法搜索最优点，涉及到实验设计方法、元模型方法、全局优化算法等方面的研究。本文围绕自适应序列采样方法、DIRECT全局优化算法、增量元模型方法以及Pareto多目标优化方法开展了进一步的研究，具体包括以下几个方面：
1）基于RBF元模型的自适应序列采样方法。由于很难确定合适的采样点数量，一次采样构造一个元模型通常是不合理的。自适应序列采样通过逐次增加采样并使其尽量分布于最合理位置，从而克服了一次采样的缺陷，达到以少量点构造精确元模型的效果。本文利用RBF元模型表达式比较简单，易于计算采样点处的曲率的特点，提出了以采样点位置最大曲率和采样点之间最小距离的作为采样标准。新采样方法能使新增采样点分布于影响元模型精度的波峰与波谷处，达到用较少的点反映了复杂目标模型几何特征的效果。以5个常用测试函数以及弹簧和焊接梁的最优设计为例测试新采样方法效果，并与拉丁超立方采样与网格采样进行比较。实验结果表明了该方法的精确性和有效性，而且优于其他采样方法。
2）基于元模型的DIRECT全局优化算法。本文在深入分析标准DIRECT算法原理和收敛性的基础上，针对标准DIRECT算法函数估值次数多、收敛速度慢的缺点，提出基于元模型的改进DIRECT算法。该方法利用优化过程中每次迭代所产生采样点来构造近似元模型，并在其上搜索最优点，从而加快了算法的收敛速度。将改进后的DIRECT算法应用于压力容器的优化设计，并对5种常用元模型的加速效果进行比较。实验结果表明RBF元模型对提高DIRECT算法的收敛性效果最好。
3）基于增量RBF元模型方法的全局优化算法。研究了增量拉丁超立方采样方法与增量径向基函数元模型更新方法；针对基于减法和加法的增量拉丁超立方采样方法难以控制采样点数量以及新增点必须是原有采样点整数倍的缺陷，改进了增量拉丁超立方采样方法；在此基础上，结合增量RBF元模型方法提出一种全新的全局优化方法。以2种常用测试函数与焊接梁为例进行测试，实验结果证明了该方法的可行性和有效性。
4）基于元模型的多目标优化方法。文本针对现有基于元模型的多目标方法大多是单值元模型的缺点提出响应面集的概念。响应面集利用RBF元模型表达式为线性方程的特点，将原有的系数向量转换为系数矩阵，使得多个子目标函数可与同一个元模型相对应。针对大规模采样直接计算Pareto适应度困难的问题，本文提出一种可增量更新的迭代式Pareto适应度计算方法。该方法克服了采样点过多导致Pareto适应度矩阵庞大的问题，并且充分利用上一次迭代中所产生的适应度值信息进行增量更新，从而减少了计算量。将响应面集与增量Pareto适应度计算方法集成到基于元模型的多目标优化算法中，并应用于两杆对称桁架与I字衡量的优化设计，实验结果表明了该方法的工程实用性与有效性。
最后，对本文的主要研究成果和创新点进行了回顾和总结，并对基于元模型的全局优化下一步研究热点和未来发展方向进行了探讨。
Mordern mechanical and electrical products become more and more complex, and it need more and more computing resource to simulate and analysis these products. Although computer technology developed rapidly, and the speed of computer become faster, it hard to meet the industry demand for simulation and analysis. It reported that an analysis of simulating automobile collision need36-160hours to simulate, and an optimization of2variables will cost almost from75days to11months. It is hard to acceptable in practical engineering applications. To deal with the challenges, the metamodel method was widely used in industry product design. This method could cut the simulation times of simulation without affecting the accuracy of simuation target model, and reduce the consumption of computing resources.
The method of metamodel is that replace complex and expensive simulation target model with simplified model which constructed by sampling points generated through experiment design.The global optimization based on metamodel is that use metamodel to search global optimum and it include the research of design of experiment, metamodel, global optimization. This article will discuss the adaptively sequential sampling, DIRECT algorithm, and incremental metamodel and Pareto multi-objective optimization and it include this content as follows.
1) Adaptively sequential sampling based on RBF metamodel. Because of hard to determin the number of sampling points, it is commonly unreasonable that construct metamode through one time sampling. To construct metamodel with a few points and overcome the defect of one time sampling method, adaptively sequential sampling increased points and make it at reasonable position as fas as possible. In order to utilize the geometrical feature of metamodel, maximum curvature of the response surface and minimum distance among the sampling sites, as a general sampling criterion, are adopted in sequential sampling procedure. For the simplicity of RBF model, we can easily evaluate curvature on design optimum via computing the difference and Hessian matrix. A new model approximation algorithm integrated sequential optimal sampling is presented. To illustrate the efficiency and accuracy of the proposed algorithm,5benchmark function and stress spring and weld beam problem have been tested, the result show that the new method is better than latin hyper-cube design and grid sampling.
2) DIRECT algorithm based on metamodel. Based on deeply analysing the principle and convergence properties of DIRECT algorithm, to overcome the defect of DIRECT method, which is too much evaluation of objective function and slow convergence rate, a modified DIRECT method based on radial basis functions was proposed. The presented method identifies the optimum area containing the global or local optimum through analysing the information of sampling points. It enhances the convergence rate of DIRECT method through constructing the radial basis functions on the sampling points which collected from optimum area and searching global optimum on the metamodel. The presented method is applied to some numerical examples and a pressure vessel design and compare accelerated effects with5common metamodel.The results of these examples demonstrate that the best metamodel is RBF metamodel.
3) A new global optimization method based on incremental metamodel. Based on research of incremental Latin Hyper-cube design and incremental updating of RBF metamodel. A new global optimization algorithm was proposed for the complex simulation model problem. To overcome the defect of old incremental latin hyper-cube sampling, which is hard to control the number of sampling points and limited to multiples, we proposed a improved incremental latin hyper-cube sampling method based on subtraction rule ideal. Combined incremental Latin hyper-cube sampling and the method of incremental update RBF metamodel, we proposed a new efficient global optimization algorithm. The presented method was applied in2commonly test functions and a weld beam problem. The results of the example demonstrated the efficiency and engineering practicability of the presented method.
4) Multi-objective optimization based on metamodel. To overcome the defect of the old multi-objective opitimization based on metamodel which used single value metamodel, a subgoal function must construct a correspond metatmodel. We proposed the conception of response surface set; which utilize the linear feature of RBF metamodel and converte the coefficient vector to coefficient matrix, made multiple subgoals function mapping one metamodel. It was hard to directly calculate Pareto fitness of massive sampling points; In order to make full use of information of sampling points generated by last iteration and reduce the computational complexity of optimization algorithm, we proposed a new incremental updating method for calculate Pareto fitness, which overcomes the defect of large-scale matrix caused by excessive sampling points. A new multi-objective optimization algorithm based on metamodel integrated response surface set and method of incrementally calculating Pareto fitness is proposed. The presented method was applied two-bar truss design problem and I-beam design. The results of the example demonstrated the efficiency and engineering practicability of the presented method.
Finally, the main research findings and innovations have been viewed and summarized at the end the article, and the challenges and future development of global optimization based on metamodel have been discussed.
引文
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[100] Weston J, Watkins C. Multi-class Support Vector Machines：J-CSD-TR-98-043．Royal Holloway University of London，1998
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[102] Fredman J．Fast MARS．Department of Statistics Technical Report，Stanford:Laboratory for computational Statistics. l993
[103] Richardson S, Wang S, Jennings LS. A multivariate adaptive regression B-splinealgorithm(BMARS) for solving a class nolinear optimal feedback controlproblems. Automatica,2008,44(4):1149-1155
[104] He S, Prempain E, Wu QH. An improved particle swarm optimizer formechanical design optimization problems. Engineering Optimization,2004,36(5):585–605
[105]邹林君,吴义忠,毛虎平. Kriging模型的增量构造及其在全局优化中的应用.计算机辅助设计与图形学报,2011,23(4):1-7
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[110] Tappeta RV, Renaud JE. Interactive Multiobjective Optimization Design Strategyfor Decision Based Design. ASME J. Mech. Des.2001,123:205–215.
[111] Deb K, Jain S. Multi-Speed Gearbox Design Using Multi-Objective EvolutionaryAlgorithms. ASME J. Mech. Des,2003,125:609–619
[112] Li Y, Fadel G.M, Wiecek MM. Approximating Pareto curves using thehyperellipse7th AIAA/USAF/NASA/ISSMO Symposium on Mul-tidisciplinaryAnalysis and Optimization, St. Louis,1998, Paper No. AIAA-98-4961
[113] Parsopoulos KE, Tasoulis DK, Vrahatis MN. Multiobjective optimization usingparallel vector evaluated particle swarm optimization. In Proceedings of theIASTED International Conference on Artificial Intelligence and Applications(AIA2004). Innsbruck, Austria: ACTA Press.2004,2:823-828.
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