全局最优解的最优性条件及凹凸化法的研究
|
摘要
本文研究全局最优化问题,提出了全局优化问题的一些最优性条件,共分为四章。第一章介绍了全局最优化问题的历史以及研究现状,一些全局优化问题的基本定义和一般结论将在第二章中给出。第三章给出了一类新的全局最优解的条件:H -差商法。首先给出L-次梯度的概念,并据此给出H -差商和H -正规形的定义,再根据H -差商定义H -差商集,H -差商集是一些非线性函数所成的集合;然后得到关于特殊函数H -差商和H -正规形的全局最优解的充分必要条件。最后在第四章中,对于目标函数是非凸凹、非单调的非线性规划问题,给出了次正定函数的定义,并且给出了这类全局优化问题的一种新的凸凹化法。通过将目标函数直接凸化或凹化可以求得原问题的全局最优解。
Several new optimality conditions for global optimization problem are proposed in this paper which contains four chapters. Global optimization problems' history and the development are introduced in the first chapter. And then some fundamental definitions and conclusions are proposed in the second chapter. In the third chapter several new optimality conditions are studied : H-differential method. First H-differential and H-norma1 form are presented according to the L-subdifferential and then the H-differential set is also presented according to the H-differential. H-differential is a set of functions which are nonlinear functions. After that some necessary and sufficient conditions of global optimization have been obtained in terms of H-differential and H-norma1 form of special functions. In the last chapter a new method of convexification and concavification is also proposed by the definition of the subdefinite functions for the nonlinear programming problem in which the objective function is non-convex, non-concave, non-monotonous. With the objective function’s direct convexification or concavification, the global optimal solution can be easily reached.
Several new optimality conditions for global optimization problem are proposed in this paper which contains four chapters. Global optimization problems' history and the development are introduced in the first chapter. And then some fundamental definitions and conclusions are proposed in the second chapter. In the third chapter several new optimality conditions are studied : H-differential method. First H-differential and H-norma1 form are presented according to the L-subdifferential and then the H-differential set is also presented according to the H-differential. H-differential is a set of functions which are nonlinear functions. After that some necessary and sufficient conditions of global optimization have been obtained in terms of H-differential and H-norma1 form of special functions. In the last chapter a new method of convexification and concavification is also proposed by the definition of the subdefinite functions for the nonlinear programming problem in which the objective function is non-convex, non-concave, non-monotonous. With the objective function’s direct convexification or concavification, the global optimal solution can be easily reached.
引文
[1]袁亚湘,孙文瑜.最优化理论与方法[M].北京:科学出版社,2003
[2]孙文瑜,徐成贤.朱德通.最优化方法[M].北京:高等教育出版社,2004
[3] Horst R,Pardalos P M, Thoai N V,黄红选译.全局优化引论[M].北京:清华大学出版社,2003
[4] Z.Y.Wu.Sufficient global optimality conditions for weakly convex minimization problems [J]. Journal of Global Optimization,2007,39:427-440
[5]吴至友,白富生.一种新的求全局优化最优性条件的方法[J].重庆师范大学学报(自然科学版),2006,23(1):1-5
[6]翟延富,李博.全局最优化问题的下降算法[J].山东轻工业学院学报,2000,14(1):77-80
[7]李成进,孙文瑜.非凸半定规划的广义Fakars引理及最优性条件[J].高等学校计算数学学报,2008,30(2):184-192
[8]吴至友.非线性规划的单调化方法[J].重庆师范大学学报:自然科学版,2004,21(2):5—11
[9] Wu Z. Y,Zhang L.S.Convexification and Concavification Methods for some Global Optimization[J].Journal of Systems Science and Complexity,2004,17(3):421-436
[10]张连生,邬东华.非线性规划的凸化,凹化和单调化[J].数学年刊,2002,23(3):537—544
[11]何颖.一类全局优化问题的新的凸化、凹化法[J].长春大学学报,2008,18(1):1-6
[12] Wu, Z.Y.,Lee, H.W.J.,Yang, X.M..A Class of Convexification And Concavification Methods for Non-monotone Optimization problems[J].Optimization,2005b,54(6),605–625
[13] LI D.,SUN X.L.,GAO F..Convexification,Concavification in Global Optimization[J].Annals of Optimization Research,2001,105:213-226
[14] WU Z.Y.,BAI F.S.,ZHANG L.S..Convexification and Concavification for a General Class of Global Optimization Problems [J].Journal of Global Optimization,2005,31:45-60
[15]吴至友,张连生,李善良.一些类型的数学规划问题的全局最优解[J].运筹学学报,2003,7(2):9-20
[16] Alexanders,Strekalovsky.Global Optimality Conditions for Nonconvex Optimization [J]. Journal of Global Optimization,1998,12:415–434
[17] YANG X.Q..Second-Order Global Optimality Conditions for Optimization Problems[J]. Journal of Global Optimization,2004,30:271–284
[18] Pinar M.C..Sufficient Global Optimality Conditions for Bivalent Quadratic Optimization [J]. Journal of optimization theory and applications,2004,122(2):433–440
[19] Rubinov A. M.,Wu Z.Y..Optimality conditions in global optimization and their applications[J].Math. Program., Ser. B,2009,120:101–123
[20]申培萍.全局优化方法[M].北京:科学出版社,2006
[21] Hiriart-Urruty J.B..Global Optimality Conditions in Maximizing a Convex Quadratic Function Under Convex Quadratic Constraints [J].J Global Option,2001,21:445-455
[22] RoyChowdhury R,Singh Y P.Chansarkar R A, Hybridiation of gradient descent algorithms with dynamic tunneling methods for global optimization[J]. IEEE Transactions on Systems, Man and Cybernetics-Part A: Systems and Humans, 2000,30(3): 383-390.
[23] Liu Xian. A Class of Augmented Filled Functions[J].Computational Optimization and Applications, 2006, 33, 333–347.
[24]李博,曹圣山.一类全局最优化问题的动态隧道方法[J].青岛海洋大学学报,2003,33 (1):160-164
[25]李博,周伊佳.全局最优化问题的凸凹化法[J].青岛科技大学学报(自然科学版),2010, 31(3):321-324
[26]解可新,韩健,林友联.最优化方法(修订版)[M].天津:天津大学出版社,2004
[27] Jeyakumar V,Rubinov A M,Wu Z Y..Nonconvex Quadratic Minimization Problems with Quadratic Constrains: Global Optimality Conditions [J].Math Program,2007,110(3):521-541
[28]邓乃扬.无约束最优化计算方法[M].北京:科学出版社,1982
[29] YANG X Q,CHEN G Y.Class of noncenvex functions and prevariational inequalities[J].J Math Anal Appl1992,169(2):359-373
[30] AVRIEL M.Nonlincar Programming.Theory and Methods[M].New Jersey:Prentice—Hall, 1976
[31] WEIR T. Proper eficiency and duality for vector valued optimization problems[J].J Austr Math Soc Ser A,1987,43(1):21-34
[32] GEOFFRION A M.Proper eficiency and the theory of vector maximization[J].J Math Anal Appl,1968,22:618—630.
[33] WEIR,T,MOND B.Pre-invex functions in multiple objective optimization [J].J Math Anal Appl,1988,136(1):29-38
[34]江维琼.半预不变凸多目标规划的最优性条件及Wolfe型对偶定理[J].华东师范大学学报(自然科学版),2006,3:32-36
[35]徐义红,熊卫芝,汪涛.集值优化问题的Benson次梯度及其应用[J].南昌大学学报(理科版),2010,34(4):326-331
[36]席少霖.非线性最优化方法[M].北京:高等教育出版社,1992
[37]薛嘉庆.最优化原理与方法(修订版)[M].北京:冶金工业出版社,1992
[38]陈宝林.最优化理论与方法[M].北京:清华大学出版社,2005
[39] Wang Wei,Yang Yong-jian,Zhang Lian-sheng.Unification of Filled Function and Tunnelling Function in Global Optimization[J].Acta Mathematicae Applicatae Sinica,English Series, 2007,23(1):59–66
[40] Bazaraa M S,Sherali H D,Shetty C M.Nonlinear Programming[M].2nd Edition,John Wiley & Sons,New York,1993
[41] More J.Generalzations of the Trust Region Problem[J].Optim Meth Soft,1993,2:189-209
[42] Pallaschke D,Rolewicz S.Foundations of Mathematical Optimization:Convex Analysis Without Linearity[M].Dotdrechet:Kluwer Academic Publishers,1997
[43] Peng J M,Yuan Y.Optimization Conditions for the Minimization of a Quadratic with Two Quadratic Constraints[J].SIAM J Optim,1997,7(3):579-594
[44] Pinac M C.Sufficient Global Optimality Conditions for Bivalent Quadratic Optimization[J]. J Optim Theor Appl,2004,122 (2):443-450
[45] Stern R,Wolkqwiczh.Indefinite Trust Region Subproblems and Nonsymmetric Eigenvalue Perturbations[J].SIAM J Optim.,1995,5:286-313
[46] Strekalovskya.Global Optimality Conditions for Nonconvex Optimization[J].J Global Optim.,1998,12(40):415-434
[47] Zalinescuc C.Convex Ana1ysis in General Vector Spaces[M].London:World Scientific, 2002
[48] Panos M. Pardalos,H. Edwin Romerjin,Hoang Tuy.Recent developments and trends in global optimization[J].Journal of Computational and Applied Mathematics,2000,124: 209-228
[49] Wu Z. Y.,Bai F. S.,Lee H. W. J. Et.al.A filled function method for constrained global optimization [J].Journal of Global Optimization,2007,39:495–507
[50]王荣波.一类非光滑广义凸多目标规划的最优性条件[J].安徽大学学报(自然科学版), 2009,33(2),1-5
[51] Li D,Sun X Y,Biswal M P and Gao F.Convexification, concavification and monotonization In global optimization[J].Annals of operations Research,2001,10:213-226.
[52] Sun,X L. McKinnon,K and Li D.A Convexification Method for a Class of Global Optimization Problem with Application to Reliability Optimization[J].Journal of Global Optimization,2001,21:185-199
[53] Strekalovsky, A S..Global optimality conditions for nonconvex optimization[J].Glob. Optim.,1998,12: 415-434
[54] Beck A.,Teboulle M..Global optimality conditions for quadratic optimization problems with binary constraints[J].SIAM Journal on Optimization,2000,11(1):179–188
[55] DUR M,HORST R,LOCAELLI M. Necessary and Sufficient Global Optimality Conditions for Convex Maximization Revisited [J].J Math Appl,1998,217(2):637-649
[56]吕一兵,姚天祥,陈忠.求解线性二层规划的一种全局优化方法[J].长江大学学报(理工卷)2008,5(4):7-10
[57]张甲,田致远,李敬玉.一类非凸二次规划问题的全局最优性条件[J].青岛大学学报(自然科学版),2010,23(3):20-23
[58] GAO Ying,RONG Wei-dong.Optimality conditions and duality for a class of nondifferentiable multiobjective generalized fractional programming problems[J].高校应用数学学报(英文版),2008,23(3):331-344
[59]赵克全,罗杰,唐莉萍.一类非光滑规划问题的最优性条件[J].重庆师范大学学报(自然科学版),2010,27(2):1-3
[60]曹圣山,李博.一类总体极值问题的进化计算法[J].山东大学学报(理学版),2003,38 (1):52-54
[61]全靖,吴至友.单调优化的一种新的凸化、凹化方法[J].重庆师范大学学报(自然科学版),2004,21(4):10-13
[62]李博,周伊佳.全局最优化问题的一些最优性条件[J].青岛科技大学学报(自然科学版), 2010,31(2):214-216
[2]孙文瑜,徐成贤.朱德通.最优化方法[M].北京:高等教育出版社,2004
[3] Horst R,Pardalos P M, Thoai N V,黄红选译.全局优化引论[M].北京:清华大学出版社,2003
[4] Z.Y.Wu.Sufficient global optimality conditions for weakly convex minimization problems [J]. Journal of Global Optimization,2007,39:427-440
[5]吴至友,白富生.一种新的求全局优化最优性条件的方法[J].重庆师范大学学报(自然科学版),2006,23(1):1-5
[6]翟延富,李博.全局最优化问题的下降算法[J].山东轻工业学院学报,2000,14(1):77-80
[7]李成进,孙文瑜.非凸半定规划的广义Fakars引理及最优性条件[J].高等学校计算数学学报,2008,30(2):184-192
[8]吴至友.非线性规划的单调化方法[J].重庆师范大学学报:自然科学版,2004,21(2):5—11
[9] Wu Z. Y,Zhang L.S.Convexification and Concavification Methods for some Global Optimization[J].Journal of Systems Science and Complexity,2004,17(3):421-436
[10]张连生,邬东华.非线性规划的凸化,凹化和单调化[J].数学年刊,2002,23(3):537—544
[11]何颖.一类全局优化问题的新的凸化、凹化法[J].长春大学学报,2008,18(1):1-6
[12] Wu, Z.Y.,Lee, H.W.J.,Yang, X.M..A Class of Convexification And Concavification Methods for Non-monotone Optimization problems[J].Optimization,2005b,54(6),605–625
[13] LI D.,SUN X.L.,GAO F..Convexification,Concavification in Global Optimization[J].Annals of Optimization Research,2001,105:213-226
[14] WU Z.Y.,BAI F.S.,ZHANG L.S..Convexification and Concavification for a General Class of Global Optimization Problems [J].Journal of Global Optimization,2005,31:45-60
[15]吴至友,张连生,李善良.一些类型的数学规划问题的全局最优解[J].运筹学学报,2003,7(2):9-20
[16] Alexanders,Strekalovsky.Global Optimality Conditions for Nonconvex Optimization [J]. Journal of Global Optimization,1998,12:415–434
[17] YANG X.Q..Second-Order Global Optimality Conditions for Optimization Problems[J]. Journal of Global Optimization,2004,30:271–284
[18] Pinar M.C..Sufficient Global Optimality Conditions for Bivalent Quadratic Optimization [J]. Journal of optimization theory and applications,2004,122(2):433–440
[19] Rubinov A. M.,Wu Z.Y..Optimality conditions in global optimization and their applications[J].Math. Program., Ser. B,2009,120:101–123
[20]申培萍.全局优化方法[M].北京:科学出版社,2006
[21] Hiriart-Urruty J.B..Global Optimality Conditions in Maximizing a Convex Quadratic Function Under Convex Quadratic Constraints [J].J Global Option,2001,21:445-455
[22] RoyChowdhury R,Singh Y P.Chansarkar R A, Hybridiation of gradient descent algorithms with dynamic tunneling methods for global optimization[J]. IEEE Transactions on Systems, Man and Cybernetics-Part A: Systems and Humans, 2000,30(3): 383-390.
[23] Liu Xian. A Class of Augmented Filled Functions[J].Computational Optimization and Applications, 2006, 33, 333–347.
[24]李博,曹圣山.一类全局最优化问题的动态隧道方法[J].青岛海洋大学学报,2003,33 (1):160-164
[25]李博,周伊佳.全局最优化问题的凸凹化法[J].青岛科技大学学报(自然科学版),2010, 31(3):321-324
[26]解可新,韩健,林友联.最优化方法(修订版)[M].天津:天津大学出版社,2004
[27] Jeyakumar V,Rubinov A M,Wu Z Y..Nonconvex Quadratic Minimization Problems with Quadratic Constrains: Global Optimality Conditions [J].Math Program,2007,110(3):521-541
[28]邓乃扬.无约束最优化计算方法[M].北京:科学出版社,1982
[29] YANG X Q,CHEN G Y.Class of noncenvex functions and prevariational inequalities[J].J Math Anal Appl1992,169(2):359-373
[30] AVRIEL M.Nonlincar Programming.Theory and Methods[M].New Jersey:Prentice—Hall, 1976
[31] WEIR T. Proper eficiency and duality for vector valued optimization problems[J].J Austr Math Soc Ser A,1987,43(1):21-34
[32] GEOFFRION A M.Proper eficiency and the theory of vector maximization[J].J Math Anal Appl,1968,22:618—630.
[33] WEIR,T,MOND B.Pre-invex functions in multiple objective optimization [J].J Math Anal Appl,1988,136(1):29-38
[34]江维琼.半预不变凸多目标规划的最优性条件及Wolfe型对偶定理[J].华东师范大学学报(自然科学版),2006,3:32-36
[35]徐义红,熊卫芝,汪涛.集值优化问题的Benson次梯度及其应用[J].南昌大学学报(理科版),2010,34(4):326-331
[36]席少霖.非线性最优化方法[M].北京:高等教育出版社,1992
[37]薛嘉庆.最优化原理与方法(修订版)[M].北京:冶金工业出版社,1992
[38]陈宝林.最优化理论与方法[M].北京:清华大学出版社,2005
[39] Wang Wei,Yang Yong-jian,Zhang Lian-sheng.Unification of Filled Function and Tunnelling Function in Global Optimization[J].Acta Mathematicae Applicatae Sinica,English Series, 2007,23(1):59–66
[40] Bazaraa M S,Sherali H D,Shetty C M.Nonlinear Programming[M].2nd Edition,John Wiley & Sons,New York,1993
[41] More J.Generalzations of the Trust Region Problem[J].Optim Meth Soft,1993,2:189-209
[42] Pallaschke D,Rolewicz S.Foundations of Mathematical Optimization:Convex Analysis Without Linearity[M].Dotdrechet:Kluwer Academic Publishers,1997
[43] Peng J M,Yuan Y.Optimization Conditions for the Minimization of a Quadratic with Two Quadratic Constraints[J].SIAM J Optim,1997,7(3):579-594
[44] Pinac M C.Sufficient Global Optimality Conditions for Bivalent Quadratic Optimization[J]. J Optim Theor Appl,2004,122 (2):443-450
[45] Stern R,Wolkqwiczh.Indefinite Trust Region Subproblems and Nonsymmetric Eigenvalue Perturbations[J].SIAM J Optim.,1995,5:286-313
[46] Strekalovskya.Global Optimality Conditions for Nonconvex Optimization[J].J Global Optim.,1998,12(40):415-434
[47] Zalinescuc C.Convex Ana1ysis in General Vector Spaces[M].London:World Scientific, 2002
[48] Panos M. Pardalos,H. Edwin Romerjin,Hoang Tuy.Recent developments and trends in global optimization[J].Journal of Computational and Applied Mathematics,2000,124: 209-228
[49] Wu Z. Y.,Bai F. S.,Lee H. W. J. Et.al.A filled function method for constrained global optimization [J].Journal of Global Optimization,2007,39:495–507
[50]王荣波.一类非光滑广义凸多目标规划的最优性条件[J].安徽大学学报(自然科学版), 2009,33(2),1-5
[51] Li D,Sun X Y,Biswal M P and Gao F.Convexification, concavification and monotonization In global optimization[J].Annals of operations Research,2001,10:213-226.
[52] Sun,X L. McKinnon,K and Li D.A Convexification Method for a Class of Global Optimization Problem with Application to Reliability Optimization[J].Journal of Global Optimization,2001,21:185-199
[53] Strekalovsky, A S..Global optimality conditions for nonconvex optimization[J].Glob. Optim.,1998,12: 415-434
[54] Beck A.,Teboulle M..Global optimality conditions for quadratic optimization problems with binary constraints[J].SIAM Journal on Optimization,2000,11(1):179–188
[55] DUR M,HORST R,LOCAELLI M. Necessary and Sufficient Global Optimality Conditions for Convex Maximization Revisited [J].J Math Appl,1998,217(2):637-649
[56]吕一兵,姚天祥,陈忠.求解线性二层规划的一种全局优化方法[J].长江大学学报(理工卷)2008,5(4):7-10
[57]张甲,田致远,李敬玉.一类非凸二次规划问题的全局最优性条件[J].青岛大学学报(自然科学版),2010,23(3):20-23
[58] GAO Ying,RONG Wei-dong.Optimality conditions and duality for a class of nondifferentiable multiobjective generalized fractional programming problems[J].高校应用数学学报(英文版),2008,23(3):331-344
[59]赵克全,罗杰,唐莉萍.一类非光滑规划问题的最优性条件[J].重庆师范大学学报(自然科学版),2010,27(2):1-3
[60]曹圣山,李博.一类总体极值问题的进化计算法[J].山东大学学报(理学版),2003,38 (1):52-54
[61]全靖,吴至友.单调优化的一种新的凸化、凹化方法[J].重庆师范大学学报(自然科学版),2004,21(4):10-13
[62]李博,周伊佳.全局最优化问题的一些最优性条件[J].青岛科技大学学报(自然科学版), 2010,31(2):214-216