支持向量回归机代理模型设计优化及应用研究
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摘要
制造业是衡量一个国家的重要支柱产业,它的强弱将影响到综合国力的兴衰。随着各学科理论和计算机仿真技术的不断发展,现代产品的研发通常采用基于仿真的设计优化,但很多情况下,产品涉及多个不同学科领域,而且各学科的仿真模型可能非常复杂,要获得理想的优化结果需要各学科分析模型之间多次迭代才能完成,计算时间的大量耗费往往令人无法接受。同时,制造业的主要竞争目标是缩短产品设计和制造周期,最终达到降低产品开发成本目的。因此,计算复杂性是复杂产品研发中面临的一个重要问题。
代理模型技术是解决以上问题的有效途径,但目前常用的代理模型对于多变量和强非线性的优化问题,逼近的效果不是很理想。为此,本文将良好性能的机器学习模型支持向量回归机引入工程优化问题,采用支持向量回归机代理模型对复杂产品设计优化进行了深入研究,开展了以下几个方面的研究工作,并取得了相关的研究成果。
1)回顾了常用代理模型和试验设计的基本理论,指出了它们各自的优缺点与适用场合;阐述了统计学习理论,提出了支持向量回归机代理模型构建方法及详细步骤;并以2个实例验证了模型的有效性。
2)提出了基于SVR-GA的优化方法、基于SVR-PSO的优化方法和基于SVR-NSGAII的优化方法;详细阐述了这些方法的算法流程;以工程多目标优化问题实例,验证了它们的有效性和可行性。较好地解决了小样本、高维数、非线性、泛化性能、局部极小点等复杂工程优化问题。
3)研究了不确定因素对产品质量特性的影响机理,提出了多目标稳健优化的数学模型;将支持向量回归机代理模型引入稳健优化,提出了基于支持向量回归机代理模型的稳健优化方法,并详细阐述该算法流程;以典型的两杆结构优化问题对所提出方法进行验证,比较研究了不同代理模型在逼近具有不确定因素的优化模型时的性能,验证了该方法的有效性。
4)介绍了五种代表性的多学科设计优化方法,并分析了各自的优缺点。指出了目前多学科协同优化方法存在的问题,提出了基于支持向量回归机代理模型的多学科协同优化方法,建立了该算法的数学模型,并详细阐述了算法流程。以典型的耦合优化问题算例对SVR-CO方法进行验证,比较研究了SVR-CO方法、标准CO与MDF方法的优化效果,验证了该方法的有效性。
Manufacturing is a measure of a country's pillar industry, and influence national comprehensive strength. With the development of disciplinary theory and computer simulation technology, complex mechanical product typically requires extensive use of simulation-based design and analysis tools, Despite the steady and continuing growth of computing power and speed, the computational cost of complex high-fidelity engineering analyses and simulations maintains pace. The high computational expense limits, or often prohibits, the use of such codes in engineering design and multidisciplinary design optimization (MDO). Meanwhile, the manufacturing industry competitively aims at shortening the product development and manufacturing cycles and reducing product development costs. Therefore, the conflicts between computational Accuracy and efficiency are is an important issue for engineering design of complex products.
Metamodeling techniques are widely used in engineering design to address these concerns. The basic approach is to construct approximations of the analysis codes that are more efficient to run, and yield insight into the functional relationship between design variables and response. In this work, we investigate support vector regression (SVR) as a metamodel for approximating complex engineering analyses, and explores the basic theory and the key implementation technologies on metamodel based on support vector regression for engineering optimization problems. The dissertation carried out researches on the following topics and obtained the corresponding results.
1) By comparing the advantages and disadvanteges of existing kinds of popolar metamodel methodology, SVR metamodel method was proposed. By using testing functions and engineering example to make comparative research on the precision of approximate models, results show SVR metamodel method is high efficiency and precision.
2) Aiming at the optimization design problem with implicit objective performance functions, a design optimization method based on SVR metamodel and genetic algorithm (GA) is proposed, a framework based on the SVR and particle swarm optimization (PSO) for structure optimization design, and a multiobjective design optimization method based on SVR metamodel and improved Non-dominated Sorting Genetic Algorithm (NSGA-II) is proposed. The structure optimization of a microwave power divider is adopted as an example to illustrate the effectiveness of these design methods.
3) Aiming at the robust optimization with uncertainty design problem of computationally intensive simulation models, a reduced approximation model technique based on SVR is introduced in order to improve the accuracy of metamodel. A framework based on SVR and GA is presented for robust optimization problems. The performances of SVR were compared with other existing metamodels under uncertainty. The applicability of the method is demonstrated using a two-bar structure system study, the results showed that the prediction accuracy of SVR model was higher than those of others metamodels, and the proposed optimization methodology is found to be accurate and efficient for robust optimization.
4) Exiting MDO methods are reviewed, and the advantages and disadvantages of these methods are discussed and analyzed. Collaborative optimization (CO) is systematically investigated. In order to deal with complicated MDO problem, a novel MDO CO method based on SVR metamodel (SVR-CO) is proposed. By using typical coupled optimization example to make comparative research of three methods including SVR-CO, CO and Multidisciplinary Feasibility Method (MDF), SVR-CO is proven to be efficient and effective.
代理模型技术是解决以上问题的有效途径,但目前常用的代理模型对于多变量和强非线性的优化问题,逼近的效果不是很理想。为此,本文将良好性能的机器学习模型支持向量回归机引入工程优化问题,采用支持向量回归机代理模型对复杂产品设计优化进行了深入研究,开展了以下几个方面的研究工作,并取得了相关的研究成果。
1)回顾了常用代理模型和试验设计的基本理论,指出了它们各自的优缺点与适用场合;阐述了统计学习理论,提出了支持向量回归机代理模型构建方法及详细步骤;并以2个实例验证了模型的有效性。
2)提出了基于SVR-GA的优化方法、基于SVR-PSO的优化方法和基于SVR-NSGAII的优化方法;详细阐述了这些方法的算法流程;以工程多目标优化问题实例,验证了它们的有效性和可行性。较好地解决了小样本、高维数、非线性、泛化性能、局部极小点等复杂工程优化问题。
3)研究了不确定因素对产品质量特性的影响机理,提出了多目标稳健优化的数学模型;将支持向量回归机代理模型引入稳健优化,提出了基于支持向量回归机代理模型的稳健优化方法,并详细阐述该算法流程;以典型的两杆结构优化问题对所提出方法进行验证,比较研究了不同代理模型在逼近具有不确定因素的优化模型时的性能,验证了该方法的有效性。
4)介绍了五种代表性的多学科设计优化方法,并分析了各自的优缺点。指出了目前多学科协同优化方法存在的问题,提出了基于支持向量回归机代理模型的多学科协同优化方法,建立了该算法的数学模型,并详细阐述了算法流程。以典型的耦合优化问题算例对SVR-CO方法进行验证,比较研究了SVR-CO方法、标准CO与MDF方法的优化效果,验证了该方法的有效性。
Manufacturing is a measure of a country's pillar industry, and influence national comprehensive strength. With the development of disciplinary theory and computer simulation technology, complex mechanical product typically requires extensive use of simulation-based design and analysis tools, Despite the steady and continuing growth of computing power and speed, the computational cost of complex high-fidelity engineering analyses and simulations maintains pace. The high computational expense limits, or often prohibits, the use of such codes in engineering design and multidisciplinary design optimization (MDO). Meanwhile, the manufacturing industry competitively aims at shortening the product development and manufacturing cycles and reducing product development costs. Therefore, the conflicts between computational Accuracy and efficiency are is an important issue for engineering design of complex products.
Metamodeling techniques are widely used in engineering design to address these concerns. The basic approach is to construct approximations of the analysis codes that are more efficient to run, and yield insight into the functional relationship between design variables and response. In this work, we investigate support vector regression (SVR) as a metamodel for approximating complex engineering analyses, and explores the basic theory and the key implementation technologies on metamodel based on support vector regression for engineering optimization problems. The dissertation carried out researches on the following topics and obtained the corresponding results.
1) By comparing the advantages and disadvanteges of existing kinds of popolar metamodel methodology, SVR metamodel method was proposed. By using testing functions and engineering example to make comparative research on the precision of approximate models, results show SVR metamodel method is high efficiency and precision.
2) Aiming at the optimization design problem with implicit objective performance functions, a design optimization method based on SVR metamodel and genetic algorithm (GA) is proposed, a framework based on the SVR and particle swarm optimization (PSO) for structure optimization design, and a multiobjective design optimization method based on SVR metamodel and improved Non-dominated Sorting Genetic Algorithm (NSGA-II) is proposed. The structure optimization of a microwave power divider is adopted as an example to illustrate the effectiveness of these design methods.
3) Aiming at the robust optimization with uncertainty design problem of computationally intensive simulation models, a reduced approximation model technique based on SVR is introduced in order to improve the accuracy of metamodel. A framework based on SVR and GA is presented for robust optimization problems. The performances of SVR were compared with other existing metamodels under uncertainty. The applicability of the method is demonstrated using a two-bar structure system study, the results showed that the prediction accuracy of SVR model was higher than those of others metamodels, and the proposed optimization methodology is found to be accurate and efficient for robust optimization.
4) Exiting MDO methods are reviewed, and the advantages and disadvantages of these methods are discussed and analyzed. Collaborative optimization (CO) is systematically investigated. In order to deal with complicated MDO problem, a novel MDO CO method based on SVR metamodel (SVR-CO) is proposed. By using typical coupled optimization example to make comparative research of three methods including SVR-CO, CO and Multidisciplinary Feasibility Method (MDF), SVR-CO is proven to be efficient and effective.
引文
[1] L. Gu. A comparison of polynomial based regression models in vehicle safety analysis. Proceedings 2001 AMSE Design Engineering Technical Conferences-Design Automation Conference, A. Diaz, ed., ASME, Pittsburgh, PA, 2001
[2] J. P. C. Kleijnen. Statisticall tools for simulation practitioners. New York: Academic Press, 1987
[3] G. E. Box, N. R. Draper. Empirical model-building and response surfaces. Wiley, New York, 1987
[4] R. H. Myers, D. C. Montgomery. Response surface methodology: Process and Product Optimization Using Designed Experiments. John Wiley & Sons Inc, New York, 1995
[5] O. Golovidov, W. H. Mason. Response surface approximations for aerodynamic parameters in high speed civil transport optimization. Technical Report, 1997
[6] S. Hosder, L. T. Watson, B. Grossman, W. H. Mason, H. Kim, R. T. Haftka, S. E. Cox. Polynomial response surface approximations for the multidisciplinary design optimization of high speed civil transport. Optim. Eng., 2001, 2(4):431-452
[7] R. Unal, R. A. Lepseh, and M. Mcmillin. Response surfaee model building and multidisciplinary optimization using D-optimal designs. Collection of Technical Papers for 7th Armual AIAA/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 1998, 18:405-411
[8] D. L. Knill, A. A. Giunta, C. A. Baker, B. Grossman, W. H. Mason, R. T. Haftka, and L. T. Watson. Response surface models combining linear and Euler aerodynamics for supersonic transport design. Aircraft, 1999, 36(1):75-86
[9] V. O. Balabanov, A. A. Giunta, O. Golovidov, B. Grossman, W. H. Mason, L. T. Watson, R. T. Haftka. Reasonable design space approach to response surface approximation. J. Aircr., 1999, 36(1):308-315
[10] J. E. Renaud, G. A. Gabriele. Improved coordination in non-hierarchic system optimization. AIAA J., 1993, 31(12):2367-2373
[11] I. P. Sobieski, I. M. Kroo. Collaborative optimization using response surface estimation. AIAA J., 2000, 38(10):1931-1938
[12] T. W. Simpson, J. D. Peplinski, P. N. Koch, J. K. Allen. Metamodels for computer-based engineering design: Survey and recommendations. Eng. Comput., 2001,17:129-150
[13] J. Sacks, W. J. Welch, T. J. Mitehell. Design and analysis of computer experiments. Statistical Science. 1989, 4(4):409-435
[14] A. A. Giunta. Aircraft multidisciplinary design optimization using design of experiments theory and response surface modeling methods. PhD dissertation, Virginia Polytechnic Institute, 1997
[15] A. J. Booker, J. E. Dennis, P. D. Frank, D. B. Serafini, V. Torczon and M. W. Trosset. A rigorous framework for optimization of expensive functions by surrogates. Structural Optimization, 1999, 17(1):1-13
[16] T. W. Simpson, T. M. Mauery, J. J. Korte and F. Mistree. Kriging metamodels for global approximation in simulation-based multidisciplinary design optimization, AIAA J., 2001, 39(12):2233-2241
[17] P. N. Koch, B. Wujek, O. Golovidov. Facilitating probabilistic multidisciplinary design optimization using Kriging approximation models. 9th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Atlanta, Georgia, 2002
[18] K. H. Lee, G. J. Park. A global robust optimization using the Kriging based approximation model. ASME International Journal, 2006, 49(3):779-788
[19] S. J. Leary, A. Bhaskar, A. J. Keane. A constraint mapping approach to the structural optimization of an expensive model using surrogates. Optim. Eng., 2001, 2:385-398
[20] S. Sakata, F. Ashida, M. Zako. Structural optimization using Kriging approximation. Computer Methods in Applied Mechanics and Engineering, 2003, 192:923-939
[21] S. Sakata, F. Ashida, M. Zako. An efficient algorithm for kriging approximation and optimization with large scale sampling data. Computer Methods in Applied Mechanics and Engineering, 2004, 193:385-404
[22]曹鸿钧,段宝岩.基于kriging模型的后优化研究.机械设计与研究,2004,20(5):10-13
[23]王晓峰,席光,王尚锦.Kriging与响应面方法在气动优化设计中的应用.工程热物理学报,2005,26(3):423-425
[24] J. D. Martin, and T. W. Simpson. Use of Kriging models to approximate deterministic computer models. AIAA J., 2005, 43(4):853-863
[25]张柱国,姚卫星,刘克龙.基于进化Kriging模型的金属加筋板结构布局优化方法.南京航空航天大学学报,2008,40(4):892-901
[26] S. N. Lophaven, H. B. Nielsen and J. S?ndergaard. DACE—A Matlab Kriging Toolbox—Version 2.0, Informatics and Mathematical Modelling, Technical University of Denmark, 2002, http://www2.imm.dtu.dk/_hbn/dace/
[27] M. Smith. Neural networks for statistical modeling. Von Nostrand Reinhold, New York, 1993
[28] M. Papadrakakis, N. D. Lagaros, Y. Tsompanakis. Structural optimization using evolution strategies and neural networks. Comput. Methods App. Mech. Eng., 1998, 156(14):309-333
[29] S. Varadarajan, W. Chen and C. Pelka. Robust concept exploration method of propulsion systems with enhanced model approximation capabilities. Engineering Optimization, 2000, 32(3):309-334
[30] Jianjiang. Chen, Renbin. Xiao, Yifang. Zhong. A response surface based hierarchical approach to multidisciplinary robust optimization design. Advanced Manufacturing Technology, 2005, 26(4):301-309
[31] C. W. Zobel, K. B. Keeling. Neural network-based simulation metamodels for predicting probability distributions. Computers and Industrial Engineering, 2008, 54(4):879-888
[32] J. Lee, H. Jeong, S. Kang. Derivative and GA-based methods in metamodeling of back-propagation neural networks for constrained approximate optimization. Structural and Multidisciplinary Optimization, 2008, 35(1):29-40
[33] B. Cheng and D. M. Titterington. Neural Networks: A review from a statistical perspective. Statistical Science, 1994, 9(1):2-54
[34]李烁,徐元铭,张俊.复合材料加筋结构的神经网络响应面优化设计.机械工程学报,2006,42(11):115-119
[35] R. L. Hardy. Multiquadratic equations of topography and other irregular surfaces. J. Geophys. Res., 1971, 76:1905-1915
[36] N. Dyn, D. Levin and S. Rippa. Numerical procedures for surface fitting of scattered data by radial basis functions. SIAM Journal of Scientific and Statistical Computing, 1986, 7(2):639-659
[37] M. Meckesheimer, R. R. Barton, T.W SimPson, F. Limayem, B. Yannou. Metamodeling of combined diserete/continuous responses. AIAA Journal , 2001, 39(10):1950-1959
[38] M. F. Hussain, R. R. Barton, S. B. Joshi. Metamodeling: Radial basis functions, versuspolynomials. Eur. J. Oper. Res., 2002, 138(1):142-154
[39] H. Nakayama, M. Arakawa, R. Sasaki. Simulation-based optimization using computational intelligence. Optim. Eng., 2002, 3(2):201-214
[40] J. H. Friedman. Multivariate adaptive regression splines. The Annals of Statistics, 1991, 19(1):1-141
[41] W. Carpenter, J. F. Barthelemy. A comparison of polynomial approximation and artificial neural nets as response surface. Technical Report 92, AIAA, 1994
[42] A. A. Giunta, L. Watson. A comparison of approximation modeling techniques: Polynomial versus interpolating models. Technical Report 98-4758, AIAA, 1998
[43] W. Shyy, P. K. Tucker, R. Vaidyanathan. Response surface and neural network techniques for rocket engine injector optimization. Technical Report 99-2455, AIAA, 1999
[44] T. W. Simpson, T. Mauery, J. Korte, F. Mistree. Comparison of response surface and Kriging models for multidisciplinary design optimization. Technical Report 98-4755, AIAA,1998
[45] R. Jin, W. Chen, T. W. Simpson. Comparative studies of metamodeling techniques under multiple modeling criteria. Struct. Multidiscip. Optim., 2001, 23(1):1-11
[46] Y. Jin. A comprehensive survey of fitness approximation in evolutionary computation. Soft Computing, 2005, 9:3-12
[47] A. Hedayat, N. Sloane, J. Stufken. Orthogonal arrays: theory and applications. Berlin: Springer, 1999
[48] M. D. Morris, T. J. Mitchell. Exploratory designs for computational experiments. Journal of Statistical Planning and Inference, 1995, 43(3):381-402
[49] R. T. Haftka, E. P. Scott, J. R. Cruz. Optimization and experiments: A survey. Applied Mechanics Review, 1998, 51(7):435-448
[50] T. H. Lee, J. J. Jung. A sampling technique enhancing accuracy and efficiency of metamodel-based RBDO: Constraint boundary sampling. Comput. Struct., 2007, 6:1-14
[51] M. H. Choueiki, C. A. Mount-Campbell. Training data development with the D-optimality criterion. IEEE Transactions on Neural Networks, 1999, 10(1):56-63
[52] Y. Freund. Boosting a weak learning algorithm by majority. Information and Computation, 1995, 121(2):256-285
[53] A. Ratle. Optimal sampling strategies for learning a fitness model. Proceedings of Congress on Evolutionary Computation, Washington DC, 1999, 3:2078-2085
[54] Y. Lin. An efficient robust concept exploration method and sequential exploratoryexperimental design. Ph.D. dissertation thesis, Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 2004
[55] R. Jin, W. Chen and A. Sudjianto. An efficient algorithm for constructing optimal design of computer experiments. J. Stat. Plan. Infer., 2005, 134(1):268-287
[56] M. Sasena, M. Parkinson, P. Goovaerts, P. Papalambros and M. Reed. Adaptive experimental design applied to an ergonomics testing procedure. Proceedings ASME 2002 Design Engineering Technical Conferences and Computer and Information in Engineering Conference, Montreal, Canada, 2002
[57] G. G. Wang. Adaptive response surface method using inherited Latin Hypercube design points. J. Mech. Des., 2003, 125:210-220
[58] G. G. Wang and T. W. Simpson. Fuzzy clustering based hierarchical metamodeling for space reduction and design optimization. Eng. Optimiz., 2004, 36(3):313-335
[59] R. Jin, W. Chen and A. Sudjianto. On sequential sampling for global metamodeling for in engineering design. Proceedings ASME 2002 Design Engineering Technical Conferences and Computer and Information in Engineering Conference, Montreal, Canada, 2002
[60] G. E. P. Box and N. R. Draper. Evolutionary operation: A statistical method for process management, Wiley, New York, 1969
[61] W. J. Welch, R. J. Buck, J. Sacks, H. P. Wynn, T. J. Mitchell and M. D. Morris. Screening, predicting and computer experiments. Technometrics, 1992, 34(1):15-25
[62] V. O. Balabanov, A. A. Giunta, O. Golovidov, B. Grossman, W. H. Mason, and L. T. Watson. Reasonable design space approach to response surface approximation. J. Aircr., 1999, 36(1):308-315
[63] V. Toropov, F. Keulen, V. Markine and H. Doer. Refinements in the multi-point approximation method to reduce the effects of noisy structural responses. Proceedings 6th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Bellevue, WA, 1996
[64] N. Alexandrov, J. E. J. Dennis, R. M. Lewis and V. Torczon. A trust region framework for managing the use of approximation models in optimization. Struct. Optim., 1998, 15(1):16-23
[65] G. Wang, S. Shan. Review of metamodeling techniques in support of engineering design optimization. Journal of Mechanical Design, 2007, 129(4):370-380
[66] B. Boser, I. Guyon, V.N. Vapnik. A training algorithm for optimal margin classifiers. Proceedings of 5th Annual Workshop on Computational Learning Theory, San Mateo, CA,1992
[67] V. N. Vapnik. Statistical learning theory. Wiley, 1998
[68] A. J. Smola, N. Murata and B. Scholkopf. A tutorial on support vector regression. NeuroCOLT Technical Report NC-TR-98-030, Royal Holloway College, University of London, UK, 1998
[69] A. J. Smola. Learning with kernels. PhD Thesis, Technical University of Berlin, 1998
[70] J. A. K. Suykens, J. Vandewalle. Least squares support vector machines classifiers, Neural Processing Letters, 1999, 9(3):293-300
[71] H. G. Chew, D. J. Crisp, R. E. Bogner, C. C. Lim. Target detecting in rader imagery using support vector machines with training size biasing. Proceedings of the 6th International Conference on Control, Automation, Robotics and Vision, 2000
[72] B. Scholkopf, A. J. Smola, R. Williamson, P. Bartlett. New support vector algorithms. Neural Computation, 2000, 12(5):1207-1245
[73] C. F. Lin, S. D. Wang. Fuzzy support vector machines. IEEE Transactions on Neural Networks, 2002, 13(2):464-471
[74] L. Zhang, W. D. Zhou, L. C. Jiao. Wavelet support vector machine. IEEE Transactions on Systems, Man and Cybernetics, Part B, 2004,34(1): 34-39
[75] C. Cortes, V. N. Vapnik. Support vector networks. Machine Learning, 1995, 20(3):273-297
[76] E. Osuna, R. Freund, F. Girosi. An improved training algorithm for support vector machines. Proceedings of the 1997 IEEE Workshop on Neural Network for Signal Processing VII, Amelea Island, 1997:276-285
[77] T. Joachims. Making large-scale SVM learning practical. Advances in Kernel Methods: Support Vector Learning. Cambridge, MA: MIT press, 1998
[78] J. C. Platt. Using analytic QP and sparseness to speed training support vector machines. Advances in Neural Information Processing Systems, MIT press, 1999:557-563
[79] S. S. Keerthi, S. Shevade, C. Bhattacharyya. Improvements to Platt’s SMO algorithm for SVM classifier design. Neural Computation, 2001, 13(3):637-649
[80] C. C. Chang, C. J. Lin. LIBSVM: A library for support vector machines(EB/OL). 2001, http://www.csie.ntu.edu.tw/~cjlin/libsvm
[81] N. Ahmed, H. Syed, K. Liu. Incremental learning with support vector machines. Proceedings of International Joint Conferneces on Artificial Intelligence, Stockholm, Sweden, 1999
[82] G. Cauwenberghs, T. Poggio. Incremental and decremental support vector machine learning.Advances in Neural Information Processing Systems, Denver, 2000:409-415
[83] M. Martin. On-line support vector machines for function approximation(EB/OL). 2002, http://www.lsi.upc.es/~mmartin/svmr.html/
[84]张学工.关于统计学习理论与支持向量机.自动化学报,2000,26(1):32-43
[85] E. Osuna, R. Freund, G. Girosi. Training support vector machines: An application to face detection. Proc. IEEE Conf. On Computer Vision and Pattern Recognition, 1997: 130-136
[86] S. Fine, J. Navratil, R. A. Gopinath. A hybrid GMM/SVM approach to speaker identification. Proccedings of IEEE International Conference on Acoustics, Speechl and Signal Proeessing, 2001, l:417-420
[87] L. J. Cao. Support vector machines experts for time series forecasting. Neuro Computing, 2003, 51:321-339
[88]张莉,席裕庚.基于支持向量机的可分离非线性动态系统辨识.自动化学报,2005,31(6):965-969
[89] K. S. Ni, T. Q. Nguyen. Image superresolution using support vector regression. IEEE Trans. Image Processing, 2007, 16(6):1596-1610
[90] J. E. Hurtado. An examination of methods for approximating implicit limit state functions from the viewpoint of statistical learning theory. Structural Safety, 2004, 26:271-293
[91] S. M. Clarke, J. H. Griebsch, T. W. Simpson. Analysis of support vector regression for approximation of complex engineering analyses. Journal of Mechanical Design, 2005, 127:1077-1087
[92] R. G. Ayestaran, F. L. Heras. Support vector regression for the design of array antennas. IEEE Antennas and Wireless Propagation Letters, 2005, 4:414-416
[93] A. Saqlain, L. S. He. Support vector regression-driven multidisciplinary design optimization for multi-stage space launch vehicle considering throttling effect. 44th AIAA Aerospace Sciences Meeting, 2006, 6:4089-4102
[94] B. P. Wang, O. Divija, Y. J. Lee. Structural optimization using FEMLAB and smooth support vector regression. 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2007, 3:2568-2577
[95] M. U. D. Qazi, L. S. He, P. Mateen. Hammersley sampling and support vector regression driven launch vehicle design. Spacecraft and Rockets, 2007, 44(5):1094-1106
[96] H. Wang, E. Y. Li, G. y. Li. The least square support vector regression coupled with parallel sampling scheme metamodeling technique and application in sheet forming optimization.Materials and Design, 2008, 8:1-12
[97] A. Basudhar, S. Missoum, A. H. Sanchez. Limit state function identification using support vector machines for discontinuous responses and disjoint failure domains. Probabilistic Engineering Mechanics, 2008, 23:1-11
[98] Y. B. Lee, O. Sangyup, D. H. Choi. Design optimization using support vector regression. Mechanical Science and Technology, 2008, 22(2):213-220
[99] B. Anirban, M. Samy. Adaptive explicit decision functions for probabilistic design and optimization using support vector machines. Comput Struct, 2008, 86:1904-1917
[100]刘春涛,林志航.基于响应面和支持向量机的产品健壮设计方法.计算机辅助设计与图形学学报,2006,18(8):1174-1178
[101] D. C. Montgomery. Design and analysis of experiments. Beijing, Chinese Press of Statistics, 1998
[102] A. F. Stevension. Theory of slot in rectangular waveguide. J Appl. Phys., 1948, 19(1):24-28
[103] L. L. Allen. The theory of array antennas. Washington: MIT Lincoln Lab, 1963
[104] J. K. Hsiao. Effects of errors on the sidelobe level of a low sidelobe array antenna. Washington: naval research laboratory, 1981
[105]钟顺时.随机误差对线阵天线性能的影响.西北电讯工程学院学报,1975,12(1):77-98
[106]向广志.超低副瓣阵列天线的公差分析.现代雷达,1996,18(6):39-48
[107]王小平,曹立明.遗传算法——理论、应用与软件实现.西安:西安交通大学出版社,2001
[108] J. KENNEDY, R. C. EBERHART. Particle swarm optimization. Proccedings of International Conference on Neural Networks. Piscataway, NJ: IEEE press, 1995:1942-1948
[109] R. C. EBERHART, J. KENNEDY. A new optimizer using particle swarm theory. Proceeding of the Sixth International Symposium on Micro Machine and Human Science. Nagoya, Japan, 1995:39-43
[110] C. A. C. Coello, T. Pulido, M. S. Lechuga. Handling multiple objectives with particle swarm optimization. IEEE Trans. Evolutionary Computation, 2004, 8(3):256-279
[111] J. D. Schaffer. Multiple objective optimization with vector evaluated genetic algorithms. Proccedings of 1st International Conference on Genetic Algorithms and Their Appplication, Lawrence Erlbaum Associates, 1985, 6:93-100
[112] D. E. Goldberg. Genetic algorithms in Search, optimization and machine Learning. Massachusetts: Addison-Wesley, 1989
[113] C. M. Fonseca, P. J. Fleming. Genetic algorithms for multiobjective optimization: Formulation,discussion and generalization. Proceedings of the 5th International Conference on Genetic Algorithms, San Mateo,California, 1993:416-423
[114] N. Srinivas, K. Deb. Multiobjective optimization using nondominated sorting in genetic algorithms. Evolutionary Computation, 1994, 2(3):221-248
[115] E. Zitzler, L. Thiele. Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach. IEEE Transactions on Evolutionary Computation, 1999, 3(4):257-271
[116] A. C. Carlos, T. P. Gregorio. Multiobjective optimization using a micro-genetic algorithm. Proceedings of the Genetic and Evolutionary Computation Conference, San Francisco, California, 2001, 274-282
[117] K. Deb, A. Pratap, S. Agarwal. A fast and elitist multiobjective genetic algorithm: NSGAⅡ. IEEE Transactions on Evolutionary Computation, 2002, 6(2):182-197
[118] C. Vladimir, Y. Q. Ma. Practical selection of SVM parameters and noise estimation for SVM regression. Neural Networks, 2004, 17:113-126
[119]杜京义等.基于遗传算法的支持向量回归机参数选取.系统工程与电子技术,2006,28(9):1430-1433
[120]陈立周.稳健设计.北京:机械工业出版,1999
[121] G. Taguchi. Quality engineering through design optimization. New York: Krauss International Press, 1986
[122] W. CHEN, J. K. ALLEN, L. TSUI K, et al. A procedure for robust design. ASME Journal Mechanical Design, 1996, 118(4): 478-485
[123] K. H. LEE, G. J. PARK. Robust optimization considering tolerances of design variables. Computers and Structures, 2001, 79(1):77-86
[124] R. JIN, X. DU, W. CHEN. The use of metamodeling techniques for optimization under uncertainty. Structure and Multidisciplinary Optimization, 2003, 25(2):99-116
[125] K. H. LEE, G. J. PARK. A general robust optimization using the Kriging based approximation method. Proceedings of the 6th World Congress on Structural and Multidisciplinary Optimization, ISSMO, Rio de Janeiro, Brail, 2005:1372-1385
[126] H. G. BEYER, B. SENDHOFF. Robust optimization—A comprehensive survey. Computer Methods in Applied Mechanics and Engineering, 2007, 196(33):3190-3218
[127]王奕首,史彦军,滕弘飞.多学科设计优化研究进展.计算机集成制造系统,2005,11(6):751-756
[128] J. S. Sobieski, J. F. M. Barthelemy, K. M. Riley. Sensitivity of optimum solutions to problem parameters. AIAA Paper 81-0548R and AIAA J., 1982, 20(9):1291-1299
[129] E. J. Cramer, P. D. Frank. Formulation for multidisciplinary optimization. SIAM Journal on Optimization, 1994, 4(4):754-776
[130] R. J. Balling, J. S. Sobieski. Optimization of coupled systems: A critical overview of approaches. AIAA Journal, 1996, 34(1):6-17
[131] J. S. Sobieski. Optimization by decomposition: A step from hierarchic to non-hierarchic systems. Second NASA/Air Force Symposium on Recent Advances in Multidisciplinary Analysis and Optimization, Hampton, Virginia, 1988
[132] J. E. Renaud, G. A. Gabriele. Improved coordination in non-hierarchic system optimization. AIAA Journal, 1993, 31(12):2367-2373
[133] R. S. Sellar, S. M. Batill, J. E. Renaud. Response surface based, concurrent subspace optimization for multidisciplinary system design. 34th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada. 1996
[134] J. S. Sobieski, J. S. Agte, R. J. Sandusky. Bi-level integrated system synthesis (BLISS). 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis, Missouri, 1998
[135] S. Kodiyalam. Bilevel Integrated System Synthesis with Response Surface. AIAA Journal, 2000, 38(8):1479-1485
[136] I. Kroo, S. Altus, R. Braun, etal. Multidisciplinary optimization methods for aircraft preliminary design. 5th AIAA/USAF/N ASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Bellevue, Washington, 1996
[137] N. M. Alexandrov, R. M. Lewis. Analytical and computational aspects of collaborative optimization for multidisciplinary design. AIAA Journal, 2002, 40(2):302-309
[2] J. P. C. Kleijnen. Statisticall tools for simulation practitioners. New York: Academic Press, 1987
[3] G. E. Box, N. R. Draper. Empirical model-building and response surfaces. Wiley, New York, 1987
[4] R. H. Myers, D. C. Montgomery. Response surface methodology: Process and Product Optimization Using Designed Experiments. John Wiley & Sons Inc, New York, 1995
[5] O. Golovidov, W. H. Mason. Response surface approximations for aerodynamic parameters in high speed civil transport optimization. Technical Report, 1997
[6] S. Hosder, L. T. Watson, B. Grossman, W. H. Mason, H. Kim, R. T. Haftka, S. E. Cox. Polynomial response surface approximations for the multidisciplinary design optimization of high speed civil transport. Optim. Eng., 2001, 2(4):431-452
[7] R. Unal, R. A. Lepseh, and M. Mcmillin. Response surfaee model building and multidisciplinary optimization using D-optimal designs. Collection of Technical Papers for 7th Armual AIAA/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 1998, 18:405-411
[8] D. L. Knill, A. A. Giunta, C. A. Baker, B. Grossman, W. H. Mason, R. T. Haftka, and L. T. Watson. Response surface models combining linear and Euler aerodynamics for supersonic transport design. Aircraft, 1999, 36(1):75-86
[9] V. O. Balabanov, A. A. Giunta, O. Golovidov, B. Grossman, W. H. Mason, L. T. Watson, R. T. Haftka. Reasonable design space approach to response surface approximation. J. Aircr., 1999, 36(1):308-315
[10] J. E. Renaud, G. A. Gabriele. Improved coordination in non-hierarchic system optimization. AIAA J., 1993, 31(12):2367-2373
[11] I. P. Sobieski, I. M. Kroo. Collaborative optimization using response surface estimation. AIAA J., 2000, 38(10):1931-1938
[12] T. W. Simpson, J. D. Peplinski, P. N. Koch, J. K. Allen. Metamodels for computer-based engineering design: Survey and recommendations. Eng. Comput., 2001,17:129-150
[13] J. Sacks, W. J. Welch, T. J. Mitehell. Design and analysis of computer experiments. Statistical Science. 1989, 4(4):409-435
[14] A. A. Giunta. Aircraft multidisciplinary design optimization using design of experiments theory and response surface modeling methods. PhD dissertation, Virginia Polytechnic Institute, 1997
[15] A. J. Booker, J. E. Dennis, P. D. Frank, D. B. Serafini, V. Torczon and M. W. Trosset. A rigorous framework for optimization of expensive functions by surrogates. Structural Optimization, 1999, 17(1):1-13
[16] T. W. Simpson, T. M. Mauery, J. J. Korte and F. Mistree. Kriging metamodels for global approximation in simulation-based multidisciplinary design optimization, AIAA J., 2001, 39(12):2233-2241
[17] P. N. Koch, B. Wujek, O. Golovidov. Facilitating probabilistic multidisciplinary design optimization using Kriging approximation models. 9th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Atlanta, Georgia, 2002
[18] K. H. Lee, G. J. Park. A global robust optimization using the Kriging based approximation model. ASME International Journal, 2006, 49(3):779-788
[19] S. J. Leary, A. Bhaskar, A. J. Keane. A constraint mapping approach to the structural optimization of an expensive model using surrogates. Optim. Eng., 2001, 2:385-398
[20] S. Sakata, F. Ashida, M. Zako. Structural optimization using Kriging approximation. Computer Methods in Applied Mechanics and Engineering, 2003, 192:923-939
[21] S. Sakata, F. Ashida, M. Zako. An efficient algorithm for kriging approximation and optimization with large scale sampling data. Computer Methods in Applied Mechanics and Engineering, 2004, 193:385-404
[22]曹鸿钧,段宝岩.基于kriging模型的后优化研究.机械设计与研究,2004,20(5):10-13
[23]王晓峰,席光,王尚锦.Kriging与响应面方法在气动优化设计中的应用.工程热物理学报,2005,26(3):423-425
[24] J. D. Martin, and T. W. Simpson. Use of Kriging models to approximate deterministic computer models. AIAA J., 2005, 43(4):853-863
[25]张柱国,姚卫星,刘克龙.基于进化Kriging模型的金属加筋板结构布局优化方法.南京航空航天大学学报,2008,40(4):892-901
[26] S. N. Lophaven, H. B. Nielsen and J. S?ndergaard. DACE—A Matlab Kriging Toolbox—Version 2.0, Informatics and Mathematical Modelling, Technical University of Denmark, 2002, http://www2.imm.dtu.dk/_hbn/dace/
[27] M. Smith. Neural networks for statistical modeling. Von Nostrand Reinhold, New York, 1993
[28] M. Papadrakakis, N. D. Lagaros, Y. Tsompanakis. Structural optimization using evolution strategies and neural networks. Comput. Methods App. Mech. Eng., 1998, 156(14):309-333
[29] S. Varadarajan, W. Chen and C. Pelka. Robust concept exploration method of propulsion systems with enhanced model approximation capabilities. Engineering Optimization, 2000, 32(3):309-334
[30] Jianjiang. Chen, Renbin. Xiao, Yifang. Zhong. A response surface based hierarchical approach to multidisciplinary robust optimization design. Advanced Manufacturing Technology, 2005, 26(4):301-309
[31] C. W. Zobel, K. B. Keeling. Neural network-based simulation metamodels for predicting probability distributions. Computers and Industrial Engineering, 2008, 54(4):879-888
[32] J. Lee, H. Jeong, S. Kang. Derivative and GA-based methods in metamodeling of back-propagation neural networks for constrained approximate optimization. Structural and Multidisciplinary Optimization, 2008, 35(1):29-40
[33] B. Cheng and D. M. Titterington. Neural Networks: A review from a statistical perspective. Statistical Science, 1994, 9(1):2-54
[34]李烁,徐元铭,张俊.复合材料加筋结构的神经网络响应面优化设计.机械工程学报,2006,42(11):115-119
[35] R. L. Hardy. Multiquadratic equations of topography and other irregular surfaces. J. Geophys. Res., 1971, 76:1905-1915
[36] N. Dyn, D. Levin and S. Rippa. Numerical procedures for surface fitting of scattered data by radial basis functions. SIAM Journal of Scientific and Statistical Computing, 1986, 7(2):639-659
[37] M. Meckesheimer, R. R. Barton, T.W SimPson, F. Limayem, B. Yannou. Metamodeling of combined diserete/continuous responses. AIAA Journal , 2001, 39(10):1950-1959
[38] M. F. Hussain, R. R. Barton, S. B. Joshi. Metamodeling: Radial basis functions, versuspolynomials. Eur. J. Oper. Res., 2002, 138(1):142-154
[39] H. Nakayama, M. Arakawa, R. Sasaki. Simulation-based optimization using computational intelligence. Optim. Eng., 2002, 3(2):201-214
[40] J. H. Friedman. Multivariate adaptive regression splines. The Annals of Statistics, 1991, 19(1):1-141
[41] W. Carpenter, J. F. Barthelemy. A comparison of polynomial approximation and artificial neural nets as response surface. Technical Report 92, AIAA, 1994
[42] A. A. Giunta, L. Watson. A comparison of approximation modeling techniques: Polynomial versus interpolating models. Technical Report 98-4758, AIAA, 1998
[43] W. Shyy, P. K. Tucker, R. Vaidyanathan. Response surface and neural network techniques for rocket engine injector optimization. Technical Report 99-2455, AIAA, 1999
[44] T. W. Simpson, T. Mauery, J. Korte, F. Mistree. Comparison of response surface and Kriging models for multidisciplinary design optimization. Technical Report 98-4755, AIAA,1998
[45] R. Jin, W. Chen, T. W. Simpson. Comparative studies of metamodeling techniques under multiple modeling criteria. Struct. Multidiscip. Optim., 2001, 23(1):1-11
[46] Y. Jin. A comprehensive survey of fitness approximation in evolutionary computation. Soft Computing, 2005, 9:3-12
[47] A. Hedayat, N. Sloane, J. Stufken. Orthogonal arrays: theory and applications. Berlin: Springer, 1999
[48] M. D. Morris, T. J. Mitchell. Exploratory designs for computational experiments. Journal of Statistical Planning and Inference, 1995, 43(3):381-402
[49] R. T. Haftka, E. P. Scott, J. R. Cruz. Optimization and experiments: A survey. Applied Mechanics Review, 1998, 51(7):435-448
[50] T. H. Lee, J. J. Jung. A sampling technique enhancing accuracy and efficiency of metamodel-based RBDO: Constraint boundary sampling. Comput. Struct., 2007, 6:1-14
[51] M. H. Choueiki, C. A. Mount-Campbell. Training data development with the D-optimality criterion. IEEE Transactions on Neural Networks, 1999, 10(1):56-63
[52] Y. Freund. Boosting a weak learning algorithm by majority. Information and Computation, 1995, 121(2):256-285
[53] A. Ratle. Optimal sampling strategies for learning a fitness model. Proceedings of Congress on Evolutionary Computation, Washington DC, 1999, 3:2078-2085
[54] Y. Lin. An efficient robust concept exploration method and sequential exploratoryexperimental design. Ph.D. dissertation thesis, Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA, 2004
[55] R. Jin, W. Chen and A. Sudjianto. An efficient algorithm for constructing optimal design of computer experiments. J. Stat. Plan. Infer., 2005, 134(1):268-287
[56] M. Sasena, M. Parkinson, P. Goovaerts, P. Papalambros and M. Reed. Adaptive experimental design applied to an ergonomics testing procedure. Proceedings ASME 2002 Design Engineering Technical Conferences and Computer and Information in Engineering Conference, Montreal, Canada, 2002
[57] G. G. Wang. Adaptive response surface method using inherited Latin Hypercube design points. J. Mech. Des., 2003, 125:210-220
[58] G. G. Wang and T. W. Simpson. Fuzzy clustering based hierarchical metamodeling for space reduction and design optimization. Eng. Optimiz., 2004, 36(3):313-335
[59] R. Jin, W. Chen and A. Sudjianto. On sequential sampling for global metamodeling for in engineering design. Proceedings ASME 2002 Design Engineering Technical Conferences and Computer and Information in Engineering Conference, Montreal, Canada, 2002
[60] G. E. P. Box and N. R. Draper. Evolutionary operation: A statistical method for process management, Wiley, New York, 1969
[61] W. J. Welch, R. J. Buck, J. Sacks, H. P. Wynn, T. J. Mitchell and M. D. Morris. Screening, predicting and computer experiments. Technometrics, 1992, 34(1):15-25
[62] V. O. Balabanov, A. A. Giunta, O. Golovidov, B. Grossman, W. H. Mason, and L. T. Watson. Reasonable design space approach to response surface approximation. J. Aircr., 1999, 36(1):308-315
[63] V. Toropov, F. Keulen, V. Markine and H. Doer. Refinements in the multi-point approximation method to reduce the effects of noisy structural responses. Proceedings 6th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Bellevue, WA, 1996
[64] N. Alexandrov, J. E. J. Dennis, R. M. Lewis and V. Torczon. A trust region framework for managing the use of approximation models in optimization. Struct. Optim., 1998, 15(1):16-23
[65] G. Wang, S. Shan. Review of metamodeling techniques in support of engineering design optimization. Journal of Mechanical Design, 2007, 129(4):370-380
[66] B. Boser, I. Guyon, V.N. Vapnik. A training algorithm for optimal margin classifiers. Proceedings of 5th Annual Workshop on Computational Learning Theory, San Mateo, CA,1992
[67] V. N. Vapnik. Statistical learning theory. Wiley, 1998
[68] A. J. Smola, N. Murata and B. Scholkopf. A tutorial on support vector regression. NeuroCOLT Technical Report NC-TR-98-030, Royal Holloway College, University of London, UK, 1998
[69] A. J. Smola. Learning with kernels. PhD Thesis, Technical University of Berlin, 1998
[70] J. A. K. Suykens, J. Vandewalle. Least squares support vector machines classifiers, Neural Processing Letters, 1999, 9(3):293-300
[71] H. G. Chew, D. J. Crisp, R. E. Bogner, C. C. Lim. Target detecting in rader imagery using support vector machines with training size biasing. Proceedings of the 6th International Conference on Control, Automation, Robotics and Vision, 2000
[72] B. Scholkopf, A. J. Smola, R. Williamson, P. Bartlett. New support vector algorithms. Neural Computation, 2000, 12(5):1207-1245
[73] C. F. Lin, S. D. Wang. Fuzzy support vector machines. IEEE Transactions on Neural Networks, 2002, 13(2):464-471
[74] L. Zhang, W. D. Zhou, L. C. Jiao. Wavelet support vector machine. IEEE Transactions on Systems, Man and Cybernetics, Part B, 2004,34(1): 34-39
[75] C. Cortes, V. N. Vapnik. Support vector networks. Machine Learning, 1995, 20(3):273-297
[76] E. Osuna, R. Freund, F. Girosi. An improved training algorithm for support vector machines. Proceedings of the 1997 IEEE Workshop on Neural Network for Signal Processing VII, Amelea Island, 1997:276-285
[77] T. Joachims. Making large-scale SVM learning practical. Advances in Kernel Methods: Support Vector Learning. Cambridge, MA: MIT press, 1998
[78] J. C. Platt. Using analytic QP and sparseness to speed training support vector machines. Advances in Neural Information Processing Systems, MIT press, 1999:557-563
[79] S. S. Keerthi, S. Shevade, C. Bhattacharyya. Improvements to Platt’s SMO algorithm for SVM classifier design. Neural Computation, 2001, 13(3):637-649
[80] C. C. Chang, C. J. Lin. LIBSVM: A library for support vector machines(EB/OL). 2001, http://www.csie.ntu.edu.tw/~cjlin/libsvm
[81] N. Ahmed, H. Syed, K. Liu. Incremental learning with support vector machines. Proceedings of International Joint Conferneces on Artificial Intelligence, Stockholm, Sweden, 1999
[82] G. Cauwenberghs, T. Poggio. Incremental and decremental support vector machine learning.Advances in Neural Information Processing Systems, Denver, 2000:409-415
[83] M. Martin. On-line support vector machines for function approximation(EB/OL). 2002, http://www.lsi.upc.es/~mmartin/svmr.html/
[84]张学工.关于统计学习理论与支持向量机.自动化学报,2000,26(1):32-43
[85] E. Osuna, R. Freund, G. Girosi. Training support vector machines: An application to face detection. Proc. IEEE Conf. On Computer Vision and Pattern Recognition, 1997: 130-136
[86] S. Fine, J. Navratil, R. A. Gopinath. A hybrid GMM/SVM approach to speaker identification. Proccedings of IEEE International Conference on Acoustics, Speechl and Signal Proeessing, 2001, l:417-420
[87] L. J. Cao. Support vector machines experts for time series forecasting. Neuro Computing, 2003, 51:321-339
[88]张莉,席裕庚.基于支持向量机的可分离非线性动态系统辨识.自动化学报,2005,31(6):965-969
[89] K. S. Ni, T. Q. Nguyen. Image superresolution using support vector regression. IEEE Trans. Image Processing, 2007, 16(6):1596-1610
[90] J. E. Hurtado. An examination of methods for approximating implicit limit state functions from the viewpoint of statistical learning theory. Structural Safety, 2004, 26:271-293
[91] S. M. Clarke, J. H. Griebsch, T. W. Simpson. Analysis of support vector regression for approximation of complex engineering analyses. Journal of Mechanical Design, 2005, 127:1077-1087
[92] R. G. Ayestaran, F. L. Heras. Support vector regression for the design of array antennas. IEEE Antennas and Wireless Propagation Letters, 2005, 4:414-416
[93] A. Saqlain, L. S. He. Support vector regression-driven multidisciplinary design optimization for multi-stage space launch vehicle considering throttling effect. 44th AIAA Aerospace Sciences Meeting, 2006, 6:4089-4102
[94] B. P. Wang, O. Divija, Y. J. Lee. Structural optimization using FEMLAB and smooth support vector regression. 48th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2007, 3:2568-2577
[95] M. U. D. Qazi, L. S. He, P. Mateen. Hammersley sampling and support vector regression driven launch vehicle design. Spacecraft and Rockets, 2007, 44(5):1094-1106
[96] H. Wang, E. Y. Li, G. y. Li. The least square support vector regression coupled with parallel sampling scheme metamodeling technique and application in sheet forming optimization.Materials and Design, 2008, 8:1-12
[97] A. Basudhar, S. Missoum, A. H. Sanchez. Limit state function identification using support vector machines for discontinuous responses and disjoint failure domains. Probabilistic Engineering Mechanics, 2008, 23:1-11
[98] Y. B. Lee, O. Sangyup, D. H. Choi. Design optimization using support vector regression. Mechanical Science and Technology, 2008, 22(2):213-220
[99] B. Anirban, M. Samy. Adaptive explicit decision functions for probabilistic design and optimization using support vector machines. Comput Struct, 2008, 86:1904-1917
[100]刘春涛,林志航.基于响应面和支持向量机的产品健壮设计方法.计算机辅助设计与图形学学报,2006,18(8):1174-1178
[101] D. C. Montgomery. Design and analysis of experiments. Beijing, Chinese Press of Statistics, 1998
[102] A. F. Stevension. Theory of slot in rectangular waveguide. J Appl. Phys., 1948, 19(1):24-28
[103] L. L. Allen. The theory of array antennas. Washington: MIT Lincoln Lab, 1963
[104] J. K. Hsiao. Effects of errors on the sidelobe level of a low sidelobe array antenna. Washington: naval research laboratory, 1981
[105]钟顺时.随机误差对线阵天线性能的影响.西北电讯工程学院学报,1975,12(1):77-98
[106]向广志.超低副瓣阵列天线的公差分析.现代雷达,1996,18(6):39-48
[107]王小平,曹立明.遗传算法——理论、应用与软件实现.西安:西安交通大学出版社,2001
[108] J. KENNEDY, R. C. EBERHART. Particle swarm optimization. Proccedings of International Conference on Neural Networks. Piscataway, NJ: IEEE press, 1995:1942-1948
[109] R. C. EBERHART, J. KENNEDY. A new optimizer using particle swarm theory. Proceeding of the Sixth International Symposium on Micro Machine and Human Science. Nagoya, Japan, 1995:39-43
[110] C. A. C. Coello, T. Pulido, M. S. Lechuga. Handling multiple objectives with particle swarm optimization. IEEE Trans. Evolutionary Computation, 2004, 8(3):256-279
[111] J. D. Schaffer. Multiple objective optimization with vector evaluated genetic algorithms. Proccedings of 1st International Conference on Genetic Algorithms and Their Appplication, Lawrence Erlbaum Associates, 1985, 6:93-100
[112] D. E. Goldberg. Genetic algorithms in Search, optimization and machine Learning. Massachusetts: Addison-Wesley, 1989
[113] C. M. Fonseca, P. J. Fleming. Genetic algorithms for multiobjective optimization: Formulation,discussion and generalization. Proceedings of the 5th International Conference on Genetic Algorithms, San Mateo,California, 1993:416-423
[114] N. Srinivas, K. Deb. Multiobjective optimization using nondominated sorting in genetic algorithms. Evolutionary Computation, 1994, 2(3):221-248
[115] E. Zitzler, L. Thiele. Multiobjective evolutionary algorithms: A comparative case study and the strength Pareto approach. IEEE Transactions on Evolutionary Computation, 1999, 3(4):257-271
[116] A. C. Carlos, T. P. Gregorio. Multiobjective optimization using a micro-genetic algorithm. Proceedings of the Genetic and Evolutionary Computation Conference, San Francisco, California, 2001, 274-282
[117] K. Deb, A. Pratap, S. Agarwal. A fast and elitist multiobjective genetic algorithm: NSGAⅡ. IEEE Transactions on Evolutionary Computation, 2002, 6(2):182-197
[118] C. Vladimir, Y. Q. Ma. Practical selection of SVM parameters and noise estimation for SVM regression. Neural Networks, 2004, 17:113-126
[119]杜京义等.基于遗传算法的支持向量回归机参数选取.系统工程与电子技术,2006,28(9):1430-1433
[120]陈立周.稳健设计.北京:机械工业出版,1999
[121] G. Taguchi. Quality engineering through design optimization. New York: Krauss International Press, 1986
[122] W. CHEN, J. K. ALLEN, L. TSUI K, et al. A procedure for robust design. ASME Journal Mechanical Design, 1996, 118(4): 478-485
[123] K. H. LEE, G. J. PARK. Robust optimization considering tolerances of design variables. Computers and Structures, 2001, 79(1):77-86
[124] R. JIN, X. DU, W. CHEN. The use of metamodeling techniques for optimization under uncertainty. Structure and Multidisciplinary Optimization, 2003, 25(2):99-116
[125] K. H. LEE, G. J. PARK. A general robust optimization using the Kriging based approximation method. Proceedings of the 6th World Congress on Structural and Multidisciplinary Optimization, ISSMO, Rio de Janeiro, Brail, 2005:1372-1385
[126] H. G. BEYER, B. SENDHOFF. Robust optimization—A comprehensive survey. Computer Methods in Applied Mechanics and Engineering, 2007, 196(33):3190-3218
[127]王奕首,史彦军,滕弘飞.多学科设计优化研究进展.计算机集成制造系统,2005,11(6):751-756
[128] J. S. Sobieski, J. F. M. Barthelemy, K. M. Riley. Sensitivity of optimum solutions to problem parameters. AIAA Paper 81-0548R and AIAA J., 1982, 20(9):1291-1299
[129] E. J. Cramer, P. D. Frank. Formulation for multidisciplinary optimization. SIAM Journal on Optimization, 1994, 4(4):754-776
[130] R. J. Balling, J. S. Sobieski. Optimization of coupled systems: A critical overview of approaches. AIAA Journal, 1996, 34(1):6-17
[131] J. S. Sobieski. Optimization by decomposition: A step from hierarchic to non-hierarchic systems. Second NASA/Air Force Symposium on Recent Advances in Multidisciplinary Analysis and Optimization, Hampton, Virginia, 1988
[132] J. E. Renaud, G. A. Gabriele. Improved coordination in non-hierarchic system optimization. AIAA Journal, 1993, 31(12):2367-2373
[133] R. S. Sellar, S. M. Batill, J. E. Renaud. Response surface based, concurrent subspace optimization for multidisciplinary system design. 34th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada. 1996
[134] J. S. Sobieski, J. S. Agte, R. J. Sandusky. Bi-level integrated system synthesis (BLISS). 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, St. Louis, Missouri, 1998
[135] S. Kodiyalam. Bilevel Integrated System Synthesis with Response Surface. AIAA Journal, 2000, 38(8):1479-1485
[136] I. Kroo, S. Altus, R. Braun, etal. Multidisciplinary optimization methods for aircraft preliminary design. 5th AIAA/USAF/N ASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, Bellevue, Washington, 1996
[137] N. M. Alexandrov, R. M. Lewis. Analytical and computational aspects of collaborative optimization for multidisciplinary design. AIAA Journal, 2002, 40(2):302-309