集值映射的高阶导数在向量优化中的应用
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摘要
本文研究了集值优化问题的高阶最优性条件、非凸集值优化问题的高阶最优性条件、约束集值优化问题的高阶Mond-Weir型及Wolfe型对偶问题、向量优化问题的高阶灵敏性和二阶稳定性。全文共分八章,具体内容如下:
在第一章里,我们介绍了集值映射向量优化问题的最优性条件和对偶理论研究、向量优化问题的稳定性和灵敏性理论研究的概况,并且阐述了本文选题的目的和主要研究工作。
在第二章里,我们介绍了本文的一些基本假设和概念。引入一个集合的广义高阶相依集和广义高阶邻接集的概念,并讨论了它们的一些性质。
在第三章里,引入集值映射的广义高阶相依上图导数和广义高阶邻接上图导数,同时讨论了这两种高阶导数的一些重要性质。基于广义高阶上图导数和Henig真有效性,获得了约束集合由一个固定集合决定的集值优化问题的高阶必要和充分最优性条件。同时还获得了约束集合由一个集值映射决定的集值优化问题的高阶Kuhn-Tucker型必要和充分最优性条件。
在第四章里,引入集值映射的高阶广义相依导数和高阶广义邻接导数,同时讨论了它们的一些性质。利用这些性质和第三章中的高阶上图导数的一些性质,在没有任何凸性假设条件下,分别研究了无约束集值优化问题和约束集值优化问题在弱有效解意义下的必要和充分最优性条件。
在第五章里,引入集值映射的高阶弱广义相依上图导数和高阶弱广义邻接上图导数,并讨论了它们的一些性质.基于高阶弱广义邻接上图导数,针对约束集值优化问题,提出了一个高阶Mond-Weir型对偶问题和一个高阶Wolfe型对偶,并讨论了相应的弱对偶、强对偶和逆对偶性质。
在第六章里,讨论的是向量优化问题的高阶灵敏性。首先,我们分别讨论了集值映射与它的剖面映射的高阶相依导数(高阶邻接导数)之间的关系。其次,给定一簇参数向量优化问题,定义了此问题的扰动映射和弱扰动映射。最后,我们讨论了弱扰动映射的高阶邻接导数与可行集映射的高阶邻接导数的弱极小点集之间的关系,同时,也讨论了扰动映射的高阶邻接导数与目标空间中的可行集映射的高阶邻接导数的极小点集和弱极小点集之间的关系。
在第七章里,讨论向量优化问题的二阶导数的稳定性。首先,我们建立了集值映射二阶相依导数和二阶邻接导数的连续性和闭性。其次,给定一簇参数向量优化问题,我们定义了此问题的弱扰动映射。最后,在恰当的假设条件下,获得了弱扰动映射的二阶邻接导数的上半连续和下半连续性。
在第八章里,我们对本文作了一个简要的总结和讨论。
In this thesis, we study higher-order optimality conditions for set-valued optimization problems, higher-order optimality conditions for nonconvex set-valued optimization problems, and higher-order Mond-Weir and Wolfe type duality problems for constrained set-valued optimization problems. We also study higher-order sensitivity and second-order stability for vector optimization problems. This thesis is divided into eight chapters. It is organized as follows:
In Chapter 1, we describe the development and current researches on the topic of optimality conditions and duality for set-valued vector optimization problems, and stability and sensitivity for vector optimization problems. We also give the motivation and the main research works.
In Chapter 2,we introduce some basic assumptions and concepts in this thesis. We introduce generalized higher-order contingent sets and generalized higher-order adjacent sets for sets, and also discuss some properties of them.
In Chapter 3, we introduce generalized higher-order contingent epiderivatives and generalized higher-order adjacent epiderivatives of set-valued maps, and discuss some important properties of the two kinds of higher-order derivatives. By virtue of generalized higher-order epiderivatives and Henig efficiency, we obtain higher-order necessary and sufficient optimality conditions for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a fixed set. Simultaneously, we also obtain higher-order Kuhn-Tucker type necessary and sufficient optimality conditions for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map.
In Chapter 4, we introduce higher-order generalized contingent derivatives and higher-order generalized adjacent derivatives for set-valued maps, and discuss some of their properties. By virtue of these properties and some properties of generalized higher-order epiderivatives in Chapter 3, without any convexity assumptions, we obtain necessary and sufficient optimality conditions of weak efficient solutions for unconstrained set-valued optimization problems and constrained set-valued optimization problems, respectively.
In Chapter 5, we introduce higher-order weakly generalized contingent epiderivatives and higher-order weakly generalized adjacent epiderivatives of set-valued maps,and discuss some of their properties. By virtue of higher-order weakly generalized adjacent epiderivatives,we introduce a higher-order Mond-Weir type dual problem and a higher-order Wolfe type dual problem for a constrained set-valued optimization problem, and discuss the corresponding weak duality, strong duality and converse duality properties, respectively.
In Chapter 6, we study higher-order sensitivity in vector optimization problem. Firstly, we discuss relationships between higher-order contingent derivatives (higher-order adjacent derivatives) of set-valued maps and their profile maps. Secondly, given a family of parametrized vector optimization problems, we define a perturbation map and a weak perturbation map for the problems, respectively. Finally, we investigate the relationship between higher-order adjacent derivatives for the weak perturbation maps and weakly minimal point sets for higher-order adjacent derivatives of feasible set maps in the objective space. We also investigate the relationships between higher-order adjacent derivatives for the perturbation maps and two types of minimal point sets (i.e. minimal point sets and weakly minimal point sets) for higher-order adjacent derivatives of feasible set maps in the objective space.
In Chapter 7, we study stability of second-order derivatives in vector optimization problems. Firstly, we establish continuity and closedness of second-order contingent derivatives and second-order adjacent derivatives for set-valued maps. Secondly, given a family of parametrized vector optimization problems, we define the weak perturbation maps for the problems. Finally, under suitable assumptions, we obtain the upper semicontinuity and the lower semicontinuity of second-order adjacent derivatives for the weak perturbation maps.
In Chapter 8,we summarize the results of this thesis and make some discussions.
在第一章里,我们介绍了集值映射向量优化问题的最优性条件和对偶理论研究、向量优化问题的稳定性和灵敏性理论研究的概况,并且阐述了本文选题的目的和主要研究工作。
在第二章里,我们介绍了本文的一些基本假设和概念。引入一个集合的广义高阶相依集和广义高阶邻接集的概念,并讨论了它们的一些性质。
在第三章里,引入集值映射的广义高阶相依上图导数和广义高阶邻接上图导数,同时讨论了这两种高阶导数的一些重要性质。基于广义高阶上图导数和Henig真有效性,获得了约束集合由一个固定集合决定的集值优化问题的高阶必要和充分最优性条件。同时还获得了约束集合由一个集值映射决定的集值优化问题的高阶Kuhn-Tucker型必要和充分最优性条件。
在第四章里,引入集值映射的高阶广义相依导数和高阶广义邻接导数,同时讨论了它们的一些性质。利用这些性质和第三章中的高阶上图导数的一些性质,在没有任何凸性假设条件下,分别研究了无约束集值优化问题和约束集值优化问题在弱有效解意义下的必要和充分最优性条件。
在第五章里,引入集值映射的高阶弱广义相依上图导数和高阶弱广义邻接上图导数,并讨论了它们的一些性质.基于高阶弱广义邻接上图导数,针对约束集值优化问题,提出了一个高阶Mond-Weir型对偶问题和一个高阶Wolfe型对偶,并讨论了相应的弱对偶、强对偶和逆对偶性质。
在第六章里,讨论的是向量优化问题的高阶灵敏性。首先,我们分别讨论了集值映射与它的剖面映射的高阶相依导数(高阶邻接导数)之间的关系。其次,给定一簇参数向量优化问题,定义了此问题的扰动映射和弱扰动映射。最后,我们讨论了弱扰动映射的高阶邻接导数与可行集映射的高阶邻接导数的弱极小点集之间的关系,同时,也讨论了扰动映射的高阶邻接导数与目标空间中的可行集映射的高阶邻接导数的极小点集和弱极小点集之间的关系。
在第七章里,讨论向量优化问题的二阶导数的稳定性。首先,我们建立了集值映射二阶相依导数和二阶邻接导数的连续性和闭性。其次,给定一簇参数向量优化问题,我们定义了此问题的弱扰动映射。最后,在恰当的假设条件下,获得了弱扰动映射的二阶邻接导数的上半连续和下半连续性。
在第八章里,我们对本文作了一个简要的总结和讨论。
In this thesis, we study higher-order optimality conditions for set-valued optimization problems, higher-order optimality conditions for nonconvex set-valued optimization problems, and higher-order Mond-Weir and Wolfe type duality problems for constrained set-valued optimization problems. We also study higher-order sensitivity and second-order stability for vector optimization problems. This thesis is divided into eight chapters. It is organized as follows:
In Chapter 1, we describe the development and current researches on the topic of optimality conditions and duality for set-valued vector optimization problems, and stability and sensitivity for vector optimization problems. We also give the motivation and the main research works.
In Chapter 2,we introduce some basic assumptions and concepts in this thesis. We introduce generalized higher-order contingent sets and generalized higher-order adjacent sets for sets, and also discuss some properties of them.
In Chapter 3, we introduce generalized higher-order contingent epiderivatives and generalized higher-order adjacent epiderivatives of set-valued maps, and discuss some important properties of the two kinds of higher-order derivatives. By virtue of generalized higher-order epiderivatives and Henig efficiency, we obtain higher-order necessary and sufficient optimality conditions for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a fixed set. Simultaneously, we also obtain higher-order Kuhn-Tucker type necessary and sufficient optimality conditions for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map.
In Chapter 4, we introduce higher-order generalized contingent derivatives and higher-order generalized adjacent derivatives for set-valued maps, and discuss some of their properties. By virtue of these properties and some properties of generalized higher-order epiderivatives in Chapter 3, without any convexity assumptions, we obtain necessary and sufficient optimality conditions of weak efficient solutions for unconstrained set-valued optimization problems and constrained set-valued optimization problems, respectively.
In Chapter 5, we introduce higher-order weakly generalized contingent epiderivatives and higher-order weakly generalized adjacent epiderivatives of set-valued maps,and discuss some of their properties. By virtue of higher-order weakly generalized adjacent epiderivatives,we introduce a higher-order Mond-Weir type dual problem and a higher-order Wolfe type dual problem for a constrained set-valued optimization problem, and discuss the corresponding weak duality, strong duality and converse duality properties, respectively.
In Chapter 6, we study higher-order sensitivity in vector optimization problem. Firstly, we discuss relationships between higher-order contingent derivatives (higher-order adjacent derivatives) of set-valued maps and their profile maps. Secondly, given a family of parametrized vector optimization problems, we define a perturbation map and a weak perturbation map for the problems, respectively. Finally, we investigate the relationship between higher-order adjacent derivatives for the weak perturbation maps and weakly minimal point sets for higher-order adjacent derivatives of feasible set maps in the objective space. We also investigate the relationships between higher-order adjacent derivatives for the perturbation maps and two types of minimal point sets (i.e. minimal point sets and weakly minimal point sets) for higher-order adjacent derivatives of feasible set maps in the objective space.
In Chapter 7, we study stability of second-order derivatives in vector optimization problems. Firstly, we establish continuity and closedness of second-order contingent derivatives and second-order adjacent derivatives for set-valued maps. Secondly, given a family of parametrized vector optimization problems, we define the weak perturbation maps for the problems. Finally, under suitable assumptions, we obtain the upper semicontinuity and the lower semicontinuity of second-order adjacent derivatives for the weak perturbation maps.
In Chapter 8,we summarize the results of this thesis and make some discussions.
引文
[1]陈光亚.优化和均衡中的某些问题[J].重庆师范大学学报(自然科学版), 2004,21(1):1-3.
[2]陈光亚.向量优化问题某些基础理论及其发展[J].重庆师范大学学报(自然科学版), 2005,22(3):6-9.
[3] J. Jahn. Vector Optimization-Theory, Applications and Extensions[M]. Springer, Berlin, 2004.
[4] D. T. Luc. Theory of Vector Optimization[M]. Springer, Berlin, 1989.
[5]林锉云,董加礼.多目标最优化理论与方法[M].长春,吉林教育出版社,1992.
[6] Y.Sawaragi, H.Nakayama, T.Tanino. Theory of Multiobjective Optimization[M]. Academic Press, New York,1985.
[7] J. P. Aubin, H. Frankowska. Set-Valued Analysis[M]. Birkh?user, Boston, 1990.
[8] G.Y.Chen, X.X.Huang, X.Q.Yang. Vector Optimization: Set-Valued and Variational Analysis [M]. Springer, Berlin, 2005.
[9]盛宝怀,刘三阳.多目标集值优化理论及其进展[J].宝鸡文理学院学报(自然科学版),2000,20(1):1-11.
[10]董加礼.不可微多目标优化[J].数学进展,1994,23(6): 517-528.
[11] W.Stadler. A survey of multicriteria optimization or the vector maximum problem, Part I: 1776-1960 [J]. Journal of Optimization Theory and Applications, 1979, 29(1): 1-52.
[12] J.P.Dauer, W.Stadler. A survey of vector optimization in infinite-dimensional spaces, Part II [J]. Journal of Optimization Theory and Applications, 1986, 51(2): 205-241.
[13] A.Chinchuluun, P.M.Pardalos. A survey of recent developments in multiobjective optimization [J]. Annals of Operations Research, 2007, 154: 29-50.
[14] J.G. Lin. Maximal vector and multi-objective optimization[J]. Journal of Optimization Theory and Applications, 1976,18(1):41-64.
[15] J. Borwein. Proper efficient points for Maximizations with respect to cones[J]. SIAM Journal of Control and Optimization, 1977,15(1):57-63.
[16]陈光亚. Banach空间中向量极值问题的Lagrange定理及Kuhn-Tucker条件[J].系统科学与数学, 1983,3(1):62-70.
[17] C. Singh. Optimality conditions in multiobjective differentiable programming[J]. Journal of Optimization Theory and Applications, 1987,53(1):115-123.
[18] T.Illes, G.Kassay. Theorems of alternative and optimality conditions for convexlike and general convexlike programming[J]. Journal of Optimization Theory and Applications, 1999, 101(2): 243-257.
[19] Z.M.Li. The optimality conditions for differentiable vector optimization problems[J]. Journal of Mathematical Analysis and Applications, 1996,201:35-43.
[20] G.Y.Chen, W.D.Rong. Characterization of the Benson proper efficiency for nonconvex vector optimization[J]. Journal of Optimization Theory and Applications, 1998,98(2):365-384.
[21] Z.M.Li. A theorem of the alternative and its application to the optimization of set-valued maps[J]. Journal of Optimization Theory and Applications, 1999, 100(2): 365-375.
[22] Z.M.Li. The optimality conditions for vector optimization of set-valued maps[J]. Journal of Mathematical Analysis and Applications, 1999, 237:413-424.
[23]李声杰. S-凸集值映射的次梯度和弱有效解[J].高校应用数学学报A辑, 1998, 13(4):463– 472.
[24] Z.F.Li, G.Y.Chen. Lagrangian multipliers, saddle points and duality in vector optimization of set-valued maps[J]. Journal of Mathematical Analysis and Applications, 1997, 215: 297- 316.
[25] X.M,Yang,X.Q.Yang,G.Y.Chen. Theorems of alternative and optimization with set-valued maps[J]. Journal of Optimization Theory and Applications, 2000, 107(3):627-643.
[26] X.M,Yang, D. Li, S.Y.Wang. Near-subconvexlikeness in vector optimization with set-valued functions[J]. Journal of Optimization Theory and Applications, 2001, 110(2):413-427.
[27] Z.F. Li. Benson proper efficiency in the vector optimization of set-valued maps[J]. Journal of Optimization Theory and Applications, 1998, 98(3):623-649.
[28] G.P. Crespi, I. Ginchev, M. Rocca. First-order optimality conditions in set-valued optimization [J]. Mathematical Methods of Operations Research, 2006, 63:87-106.
[29] Y.H.Xu, S.Y.Liu. Super efficiency in the nearly cone-subconvexlike vector optimization with set-valued functions[J]. ACTA MATHEMATICA SCIENTIA, 2005, 25B(1): 152-160.
[30] X.H. Gong. Optimality conditions for Henig and globally proper efficient solutions with order cone has empty interior[J]. Journal of Mathematical Analysis and Applications, 2005, 307: 12-37.
[31] W. D. Rong, Y. N. Wu. Characterizations of super efficiency in cone-convexlike vector optimization with set-valued maps[J]. Mathematical Methods of Operations Research, 1998,48: 247-258.
[32] C.Ling. Generalized tangent epiderivative and applications to set-valued optimization[J]. Journal of Nonlinear and Convex Analysis, 2002,3:303-313.
[33]凌晨.集值映射多目标规划的K-T最优性条件[J].系统科学与数学, 2000,20(2):196-202.
[34] R.T.Rockafellar. Proto-differentiability of set-valued mappings and its applications in optimization[J]. Annales de I’Institut Henri Poincare. Analyse Non Lineaire, 1989,6:449-482.
[35] H.W. Corley. Optimality conditions for maximizations of set-valued functions[J]. Journal ofOptimization Theory and Applications, 1988, 58(1):1-10.
[36] M. Alonso,L.Rodriguez-Marin. Optimality conditions for a nonconvex set-valued optimization problem[J]. Computers and Mathematics with Applications, 2008, 56:82- 89.
[37] J. Jahn, R. Rauh. Contingent epiderivatives and set-valued optimization[J]. Mathematical Methods of Operations Research, 1997, 46:193-211.
[38] G.Y.Chen, J.Jahn. Optimality conditions for set-valued optimization problems[J]. Mathematical Methods of Operations Research, 1998, 48:187-200.
[39] E.M.Bednarczuk, W.Song. Contingent epiderivative and its applications to set-valued optimization[J]. Control Cybernet, 27:375-386.
[40] J.Jahn, A.A. Khan. Generalized contingent epiderivatives in set-valued optimization: Optimality conditions[J]. Numerical Functional Analysis and Optimization, 2002, 23(7-8): 807-831.
[41] X. H. Gong, H. B. Dong, S. Y. Wang. Optimality conditions for proper efficient solutions of vector set-valued optimization[J]. Journal of Mathematical Analysis and Applications, 2003, 284, 332-350.
[42] D.T.Luc. Contingent derivatives of set-valued maps and applications to vector optimization[J]. Mathematical Programming ,1991, 50:99-111.
[43] W. Song. Weak subdifferential of set-valued mappings[J]. Optimization, 2003,52(3):263-276.
[44] L.J.Lin. Optimization of set-valued functions[J]. Journal of Mathematical Analysis and Applications, 1994, 186:30-51.
[45] C. S. Lalitha, R. Arora. Weak Clarks epiderivative in set-valued optimization[J]. Journal of Mathematical Analysis and Applications, 2008, 342:704-714.
[46] J. Jahn, A. A.Khan, P.Zeilinger. Second-order optimality conditions in set optimalization[J]. Journal of Optimization Theory and Applications, 2005, 125:331-347.
[47] B.H.Sheng, S.Y.Liu. On the generalized Fritz John optimality conditions of vector optimization with set-valued maps under Benson proper efficiency[J]. Applied Mathematics and Mechanics(English Edition). 2002, 23(12):1444-1451.
[48] S.J.Li,K. L.Teo,X. Q. Yang. Higher-order optimality conditions for set-valued optimization[J]. Journal of Optimization Theory and Applications, 2008, 137(3):533-553.
[49] S.J.Li,C.R.Chen. Higher-order optimality conditions for Henig efficient solutions in set- valued optimization[J]. Journal of Mathematical Analysis and Applications, 2006, 323: 1184- 1200.
[50] C.R. Chen, S. J. Li,K.L. Teo. Higher order weak epiderivatives and applications to duality and optimality conditions[J]. Computers & Mathematics with Applications, 2009, 57(8): 1389-1399.
[51] Q.L.Wang, S.J.Li. Generalized higher-order optimality conditions for set-valued optimization under Henig efficiency[J]. Numerical Functional Analysis and Optimization, 2009, 30(7-8): 849-869.
[52] Q.L.Wang, S.J.Li. Higher-order weakly generalized adjacent epiderivatives and applications to duality of set-valued optimization[J]. Journal of Inequality and Applications, 2009, Art. No462637:1-18.
[53] P.Q.Khanh, N.D.Tuan. Variational sets of multivalued mappings and a unified study of optimality conditions[J]. Journal of Optimization Theory and Applications, 2008, 139:47-65.
[54] P.Q.Khanh, N.D.Tuan. Higher-order variational sets and higher-order optimality conditions for proper efficiency in set-valued nonsmooth vector optimization[J]. Journal of Optimization Theory and Applications, 2008, 139:243-261.
[55]王其林.一类广义凸集值映射优化问题弱有效解的最优性条件[J].四川师范大学学报(自),2007,30(2):556-559.
[56] Q.L.Wang. High-order Fritz John type optimality conditions for Benson proper efficient solutions in set-valued optimization problems[J]. OR Transactions, 2009,13(3):1-9.
[57]刘三阳,盛宝怀.非凸向量集值优化Benson真有效解的最优性条件与对偶[J].应用数学学报,2003,26(2):337-344.
[58] S. J. Li, X. Q. Yang, G. Y. Chen. Nonconvex vector optimization of set-valued mappings[J]. Journal of Mathematical Analysis and Applications, 2003, 283:337-350.
[59] D.T.Luc. On duality theory in multiobjective programming[J]. Journal of Optimization Theory and Applications, 1984, 43(4):557-582.
[60] H.C. Lai, C.P.Ho. Duality theorem of nondifferentiable convex multiobjective programming[J]. Journal of Optimization Theory and Applications, 1986, 50(3):407-420.
[61]应玫茜,择一定理与多目标[J].系统科学与数学,1989,9(1):69-76.
[62] E.V. Tamminen. Sufficient conditions for the existence of multipliers and Lagrangian duality in abstract optimization problems[J]. Journal of Optimization Theory and Applications, 1994, 82(1):93-104.
[63] C.R.Beter, M.Singh. Duality for multiobjective B-invex programming involving n-set function[J]. Journal of Mathematical Analysis and Applications, 1996, 202:701-726.
[64] W. Song. Lagrangian duality for minimization of nonconvex multifunctions[J]. Journal of Optimization Theory and Applications, 1997, 93(1):297-316.
[65] R. Osuna-Gomez, A. Rufian-Lizana, P. Ruiz-Canales. Duality in nondifferetiable vector programming[J]. Journal of Mathematical Analysis and Applications, 2001, 259:462-475.
[66]王其林,李泽民.一类向量极值问题的最优性条件和Lagrange对偶[J].重庆大学学报(自),2005,28(6):106-109.
[67] T. Weir, B. Mond. Generalized convexity and duality in multiple objective programming[J]. Bulletin of the Australian Mathematical Society, 1989, 39:287-299.
[68] P. Q. Khanh. Sufficient optimality conditions and duality in vector optimization with invex-convexlike functions[J]. Journal of Optimization Theory and Applications, 1995, 87: 359-378.
[69] P. H. Sach, B. D. Craven. Invex multifunctions and duality[J]. Numerical Functional Analysis and Optimization, 1991,12:575-591.
[70] P. H. Sach, N. D. Yen, B. D. Craven. Generalized invexity and duality theorems with multifunctions[J]. Numerical Functional Analysis and Optimization, 1994, 15:131-153.
[71] S. K. Mishra, S. Y.Wang, K. K. Lai. Optimality and duality in nondifferentiable and multiobjective programming under generalized d -invexity[J]. Journal of Global Optimization, 2004, 29: 425-438.
[72] V. Preda, K.Koller. General Mond-Weir duality for multiobjectiv programming with generalized ( F ,ρ,θ)-convex set functions[J]. Romanian Journal of Pure and Application Mathematics, 2000, 45:1005-1018.
[73] S.J. Li, K.L. Teo, X. Q. Yang. Higher-order Mond-Weir duality for set-valued optimization[J]. Journal of Computational and Applied Mathematics, 2008, 217(2):339-349.
[74] A.M.Rubinov, X.Q.Yang. Lagrange type Functions in Constrained Nonconvex Optimization[M]. Kluwer Academic Publishers, Dordrecht,2003.
[75] R.T.Rockafellar, R.J.B.Wets. Variational Analysis[M]. Springer-Verlag, Berlin, 1998.
[76] C.J.Goh, X.Q.Yang. Duality in Optimization and Variational Inequalities[M]. Taylor and Francis, London, 2002.
[77] C.J.Goh, X.Q.Yang. Nonlinear Lagrangian theory for nonconvex optimization[J]. Journal of Optimization Theory and Applications, 2001, 109(1):99-121.
[78] C.Y.Wang, X.Q.Yang, X.M.Yang. Unified nonlinear Lagrangian approach to duality and optimal paths[J]. Journal of Optimization Theory and Applications, 2007,135(1):85-100.
[79] X.X.Huang, X.Q.Yang. A unified augmented Lagrangian approach to duality and exact penalization[J]. Mathematics of Operations Research, 2003, 28(3):533-552.
[80] X.Q.Yang, X.X.Huang. A ninlinear Lagrangian approachto constrained optimization problems[J]. SIAM Journal on Optimization, 2001, 11(4):1119-1144.
[81] Y.Y.Zhou, X.Q.Yang. Augmented Lagrangian function, non-quadratic growth condition and exact penalization[J]. Operations Research Letters, 2006, 34:127-134.
[82] Y.Y.Zhou, X.Q.Yang. Some results about duality and exact penalization[J]. Journal of Global Optimization, 2004, 29:497-509.
[83] A.M.Rubinov, X.X.Huang, X.Q.Yang. The zero duality gap property and lower semicontinuity of the perturbation function[J]. Mathematical Methods of Operations Research, 2002, 27(4): 775-791.
[84] W.Fenchel. On conjugate convex functions[J]. Canadian Journal of Mathematics, 1949, 1: 73- 77.
[85] J.J. Moreau. Inf-convolution des fonctions numeriques sur un espace vectoriel[J]. comptes Rendus Academie Sciences Paris, 1963, 256:5047-5049.
[86] R.T.Rockafellar. Convex Analysis[M]. Princeton University Press, Princeton,New Jersey,1970.
[87] R.T.Rockafellar. Conjugate duality and optimization[M]. CBMS-NSF Reqional Conference Series in Applied Mathematics 16, SIAM, Philadelphia, 1974.
[88] I. Ekeland, R. Temam. Convex Analysis and Variational Problems[M]. North-Holland Publishing Company, Amsterdam, Holland, 1976.
[89] T.Tanino, Y.Sawaragi. Conjugate maps and duality in multiobjective optimization[J]. Journal of Optimization Theory and Applications, 1980,31:473-499.
[90]汪寿阳.多目标最优化中的共轭对偶理论[J].系统科学与数学,1984,4(4):303-313.
[91] T.Tanino. Conjugate duality in vector optimization[J]. Journal of Mathematical Analysis and Applications, 1992,167:84-97.
[92] L. Altangerel, R.I.Bot, G. Wanka. Conjugate duality in vector optimization and some applications to the vector variational inequality[J]. Journal of Mathematical Analysis and Applications, 2007, 329:1010-1035.
[93] L. Altangerel, R.I.Bot, G. Wanka. Variational principles for vector equilibrium problems related to conjugate duality[J]. Journal of Nonlinear and Convex Analysis, 2007,8:179-196.
[94] A. Lohne. Optimization with set relations:conjugate duality[J]. Optimization, 2005, 54(3): 265-282.
[95] A. Lohne, C. Tammer. A new approach to duality in vector optimization[J]. Optimization, 2007, 56(1-2): 221-239.
[96] C.S.Lalitha, R.Arora. Conjugate maps, subgradients and conjugate duality in set-valued optimization[J]. Numerical Functional Analysis and Optimization, 2007, 28(7-8):897-909.
[97] F.Giannessi(ed.). Vector Variational Inequalities and Vector Equilibria:Mathematical Theories[C]. Kluwer Academic Publishers, Dordrecht, Holland, 2000.
[98] A.G?pfert, C.Tammer, H.Riahi, C.Zǎlinescu. Variational Methods in Partially Ordered Spaces [M]. Springer, Berlin, 2003.
[99] A.V. Fiacco. Introduction to Sensitivity and Stablity Analysis in Nonlinear Programming[M]. Academic Press, New York, 1983.
[100] R.T. Rockafellar. Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming[J]. Mathematical Programming Study, 1982,17:28-66.
[101] J.P.Aubin, I.Ekeland. Applied Nonlinear Analysis[M]. John Wiley & Sons, New York,1984.
[102] C. Berge. Topological Spaces[M]. Oliver and Boyd, London, 1963.
[103] W.W. Hogan. Point-to-set maps in mathematical programming[J]. SIAM Review, 1973, 15: 591-603.
[104] T.Tanino, Y. Aawaragi. Stability of nondominated solutions in multicriteria decision- marking[J]. Journal of Optimization Theory and Applications, 1980, 30:229-253.
[105] E. Jurkiewicz. Stability of compromise solution in multicriteria decision-making problems [J]. Journal of Optimization Theory and Applications, 1983, 40:77–83.
[106] W.Alt. Local stability of solutions to differentiable optimization problems in Banach spaces, Journal of Optimization Theory and Applications, 1991, 70(3): 443-466.
[107] G. Y. Chen, B. D. Craven. Existence and continuity of solutions for vector optimization[J]. Journal of Optimization Theory and Applications, 1994, 81:459-468.
[108] R. E. Lucchetti, E. Miglierina. Stability for convex vector optimization problems[J]. Optimization, 2004, 53:517-528.
[109] S.J. Li, G.Y. Chen, K.L. Teo. On the stability of generalized vector quasivariational inequality problems[J]. Journal of Optimization Theory and Applications, 2002, 113:283–295.
[110] Y. H. Cheng, D. L. Zhu. Global stability results for the weak vector variational inequality[J]. Journal of Global Optimization, 2005, 32:543–550.
[111] P.Q. Khanh, L.M.Luu. Upper semicontinity of the solution set to parametric vector quasivariational inequalities[J]. Journal of Global Optimization, 2005, 32:569–580.
[112] L. Q. Anh,P. Q. Khanh. On the H?lder continuity of solutions to parametric multivalued vector equilibrium problems[J]. Journal of Mathematical Analysis and Applications, 2006, 321: 308-315.
[113]傅俊义,具移动锥的参数向量均衡问题[J].南昌大学学报,2006, 30:109-111.
[114] S.J.Li, C.R.Chen. Stability of weak vector variational inequality[J]. Nonlinear Analysis:Theory, Methods & Applications, 2009, 70(4):1528-1535.
[115] C.R.Chen, S.J.Li. Semicontinuity of the solution set maps to a set-valued weak vector varivational inequality[J]. Journal of Industrial and Management Optimization, 2007, 3: 519- 528.
[116] S.J.Li, Z.M.Fang. On the stability of a dual weak vector variational inequality problem[J].Journal of Industrial and Management Optimization, 2008,4:155-165.
[117] S.W.Xiang, W.S.Yin, Stability results for efficient solutions of vector optimization problems, Journal of Optimization Theory and Applications, 2007,134:385-398.
[118] Q.L.Wang, S.J.Li. Stability analysis of second-order adjacent derivatives in multiobjective optimization[J]. 2009, submitted.
[119] T.Tanino. Sensitivity analysis in multiobjective optimization[J]. Journal of Optimization Theory and Applications, 1988, 56(3):479-499.
[120] T.Tanino, Stability and sensitivity Analysis in convex vector Optimization[J]. SIAM J. Control Optim, 1988,26:521-536.
[121] D. S. Shi. Contingent derivative of the perturbation map in multiobjective optimization[J]. Journal of Optimization Theory and Applications, 1991, 70:385-396.
[122] D. S. Shi, Sensitivity analysis in convex vector optimization[J]. Journal of Optimization Theory and Applications,1993,77(1):145-159.
[123] H. Kuk, T. Tanino, M. Tanaka. Sensitivity analysis in parametrized convex vector optimization[J]. Journal of Mathematical Analysis and Applications, 1996, 202:511-522.
[124] H. Kuk, T. Tanino, M. Tanaka, Sensitivity analysis in vector optimization[J]. Journal of Optimization Theory and Applications, 1996, 89(3):713-730.
[125] S. J. Li. Sensitivity and stability for contingent derivative in multiobjective optimization[J]. Mathematica Applicata, 1998, 11(2):49-53.
[126] W. Song, L. J. Wan. Contingent epidifferentiability of the value map in vector optimization[J]. Journal of Natural Science of Helongjiang University, 2005, 22(2):198-203.
[127] S. J. Li, H. Yan, G. Y. Chen. Differential and sensitivity properties of gap functions for vector variational inequalities[J]. Mathematical Methods of Operations Research, 2003,57:377-391.
[128] K. W. Meng, S. J. Li. Differential and sensitivity properties of gap functions for Minty vector variational inequalities[J]. Journal of Mathematical Analysis and Applications, 2008, 337: 386- 398.
[129] S. J. Li, K. W. Meng, J.P.Penot. Calculus rules for derivatives of multimaps[J]. Set-Valued and Variarional Analysis, 2009,17:21-39.
[130] M.H.Li, S. J. Li. Second-order differential and sensitivity properties of weak vector variational inequalities[J]. Journal of Optimization Theory and Applications, 2010,144(1):76-87.
[131] Q.L.Wang, S.J.Li. Higher order sensitivity analysis in non-convex vector optimization[J]. Journal of Industrial and Management Optimization, 2010, 6(2):381-392.
[132] C. R. Chen, S. J. Li. Different conjugate dual problems in vector optimization and their relations[J]. Journal of Optimization Theory and Applications, 2009, 140(3):443-461.
[133] C. R. Chen, S. J. Li. Conjugate dual problems in constrained set-valued optimization and applications[J]. European Journal of Operational Research, 2009, 196(1): 21-32.
[134] X.Q.Yang. Directional derivatives for set-valued mappings and applications[J]. Mathematical Methods of Operations Research, 1998, 48:273-285.
[135] J. M.Borwein,D. Zhuang. Supper efficiency in vector optimization[J]. Transactions of the American Mathematical Society, 1993, 338:105-122.
[136] B.Jiménez, V.Novo. Second-order necessary conditions in set constrained differentiable vector optimization[J]. Mathematical Methods of Operations Research, 2003, 58: 299-317.
[137] C.Certh, P.Weidner. Nonconvex separation theorems and some applications in vector optimization[J]. Journal of Optimization Theory and Applications, 1990, 67:297-320.
[138] R.B. Holmes. Geometric Functional Analysis and Its Applications[M]. Springer-Verlag, New York, New York, 1975.
[139] N.J.Huang, J.Li, H.B.Thompson. Stability for parametric implicit vector equilibrium problems[J]. Mathematical and Computer Modelling, 2006,43:1267-1274.
[140] X.H.Gong, J.C.Yao. Lower semicontinuity of the set of efficient solutions for generalized systems[J]. Journal of Optimization Theory and Applications, 2008,138:197-205.
[2]陈光亚.向量优化问题某些基础理论及其发展[J].重庆师范大学学报(自然科学版), 2005,22(3):6-9.
[3] J. Jahn. Vector Optimization-Theory, Applications and Extensions[M]. Springer, Berlin, 2004.
[4] D. T. Luc. Theory of Vector Optimization[M]. Springer, Berlin, 1989.
[5]林锉云,董加礼.多目标最优化理论与方法[M].长春,吉林教育出版社,1992.
[6] Y.Sawaragi, H.Nakayama, T.Tanino. Theory of Multiobjective Optimization[M]. Academic Press, New York,1985.
[7] J. P. Aubin, H. Frankowska. Set-Valued Analysis[M]. Birkh?user, Boston, 1990.
[8] G.Y.Chen, X.X.Huang, X.Q.Yang. Vector Optimization: Set-Valued and Variational Analysis [M]. Springer, Berlin, 2005.
[9]盛宝怀,刘三阳.多目标集值优化理论及其进展[J].宝鸡文理学院学报(自然科学版),2000,20(1):1-11.
[10]董加礼.不可微多目标优化[J].数学进展,1994,23(6): 517-528.
[11] W.Stadler. A survey of multicriteria optimization or the vector maximum problem, Part I: 1776-1960 [J]. Journal of Optimization Theory and Applications, 1979, 29(1): 1-52.
[12] J.P.Dauer, W.Stadler. A survey of vector optimization in infinite-dimensional spaces, Part II [J]. Journal of Optimization Theory and Applications, 1986, 51(2): 205-241.
[13] A.Chinchuluun, P.M.Pardalos. A survey of recent developments in multiobjective optimization [J]. Annals of Operations Research, 2007, 154: 29-50.
[14] J.G. Lin. Maximal vector and multi-objective optimization[J]. Journal of Optimization Theory and Applications, 1976,18(1):41-64.
[15] J. Borwein. Proper efficient points for Maximizations with respect to cones[J]. SIAM Journal of Control and Optimization, 1977,15(1):57-63.
[16]陈光亚. Banach空间中向量极值问题的Lagrange定理及Kuhn-Tucker条件[J].系统科学与数学, 1983,3(1):62-70.
[17] C. Singh. Optimality conditions in multiobjective differentiable programming[J]. Journal of Optimization Theory and Applications, 1987,53(1):115-123.
[18] T.Illes, G.Kassay. Theorems of alternative and optimality conditions for convexlike and general convexlike programming[J]. Journal of Optimization Theory and Applications, 1999, 101(2): 243-257.
[19] Z.M.Li. The optimality conditions for differentiable vector optimization problems[J]. Journal of Mathematical Analysis and Applications, 1996,201:35-43.
[20] G.Y.Chen, W.D.Rong. Characterization of the Benson proper efficiency for nonconvex vector optimization[J]. Journal of Optimization Theory and Applications, 1998,98(2):365-384.
[21] Z.M.Li. A theorem of the alternative and its application to the optimization of set-valued maps[J]. Journal of Optimization Theory and Applications, 1999, 100(2): 365-375.
[22] Z.M.Li. The optimality conditions for vector optimization of set-valued maps[J]. Journal of Mathematical Analysis and Applications, 1999, 237:413-424.
[23]李声杰. S-凸集值映射的次梯度和弱有效解[J].高校应用数学学报A辑, 1998, 13(4):463– 472.
[24] Z.F.Li, G.Y.Chen. Lagrangian multipliers, saddle points and duality in vector optimization of set-valued maps[J]. Journal of Mathematical Analysis and Applications, 1997, 215: 297- 316.
[25] X.M,Yang,X.Q.Yang,G.Y.Chen. Theorems of alternative and optimization with set-valued maps[J]. Journal of Optimization Theory and Applications, 2000, 107(3):627-643.
[26] X.M,Yang, D. Li, S.Y.Wang. Near-subconvexlikeness in vector optimization with set-valued functions[J]. Journal of Optimization Theory and Applications, 2001, 110(2):413-427.
[27] Z.F. Li. Benson proper efficiency in the vector optimization of set-valued maps[J]. Journal of Optimization Theory and Applications, 1998, 98(3):623-649.
[28] G.P. Crespi, I. Ginchev, M. Rocca. First-order optimality conditions in set-valued optimization [J]. Mathematical Methods of Operations Research, 2006, 63:87-106.
[29] Y.H.Xu, S.Y.Liu. Super efficiency in the nearly cone-subconvexlike vector optimization with set-valued functions[J]. ACTA MATHEMATICA SCIENTIA, 2005, 25B(1): 152-160.
[30] X.H. Gong. Optimality conditions for Henig and globally proper efficient solutions with order cone has empty interior[J]. Journal of Mathematical Analysis and Applications, 2005, 307: 12-37.
[31] W. D. Rong, Y. N. Wu. Characterizations of super efficiency in cone-convexlike vector optimization with set-valued maps[J]. Mathematical Methods of Operations Research, 1998,48: 247-258.
[32] C.Ling. Generalized tangent epiderivative and applications to set-valued optimization[J]. Journal of Nonlinear and Convex Analysis, 2002,3:303-313.
[33]凌晨.集值映射多目标规划的K-T最优性条件[J].系统科学与数学, 2000,20(2):196-202.
[34] R.T.Rockafellar. Proto-differentiability of set-valued mappings and its applications in optimization[J]. Annales de I’Institut Henri Poincare. Analyse Non Lineaire, 1989,6:449-482.
[35] H.W. Corley. Optimality conditions for maximizations of set-valued functions[J]. Journal ofOptimization Theory and Applications, 1988, 58(1):1-10.
[36] M. Alonso,L.Rodriguez-Marin. Optimality conditions for a nonconvex set-valued optimization problem[J]. Computers and Mathematics with Applications, 2008, 56:82- 89.
[37] J. Jahn, R. Rauh. Contingent epiderivatives and set-valued optimization[J]. Mathematical Methods of Operations Research, 1997, 46:193-211.
[38] G.Y.Chen, J.Jahn. Optimality conditions for set-valued optimization problems[J]. Mathematical Methods of Operations Research, 1998, 48:187-200.
[39] E.M.Bednarczuk, W.Song. Contingent epiderivative and its applications to set-valued optimization[J]. Control Cybernet, 27:375-386.
[40] J.Jahn, A.A. Khan. Generalized contingent epiderivatives in set-valued optimization: Optimality conditions[J]. Numerical Functional Analysis and Optimization, 2002, 23(7-8): 807-831.
[41] X. H. Gong, H. B. Dong, S. Y. Wang. Optimality conditions for proper efficient solutions of vector set-valued optimization[J]. Journal of Mathematical Analysis and Applications, 2003, 284, 332-350.
[42] D.T.Luc. Contingent derivatives of set-valued maps and applications to vector optimization[J]. Mathematical Programming ,1991, 50:99-111.
[43] W. Song. Weak subdifferential of set-valued mappings[J]. Optimization, 2003,52(3):263-276.
[44] L.J.Lin. Optimization of set-valued functions[J]. Journal of Mathematical Analysis and Applications, 1994, 186:30-51.
[45] C. S. Lalitha, R. Arora. Weak Clarks epiderivative in set-valued optimization[J]. Journal of Mathematical Analysis and Applications, 2008, 342:704-714.
[46] J. Jahn, A. A.Khan, P.Zeilinger. Second-order optimality conditions in set optimalization[J]. Journal of Optimization Theory and Applications, 2005, 125:331-347.
[47] B.H.Sheng, S.Y.Liu. On the generalized Fritz John optimality conditions of vector optimization with set-valued maps under Benson proper efficiency[J]. Applied Mathematics and Mechanics(English Edition). 2002, 23(12):1444-1451.
[48] S.J.Li,K. L.Teo,X. Q. Yang. Higher-order optimality conditions for set-valued optimization[J]. Journal of Optimization Theory and Applications, 2008, 137(3):533-553.
[49] S.J.Li,C.R.Chen. Higher-order optimality conditions for Henig efficient solutions in set- valued optimization[J]. Journal of Mathematical Analysis and Applications, 2006, 323: 1184- 1200.
[50] C.R. Chen, S. J. Li,K.L. Teo. Higher order weak epiderivatives and applications to duality and optimality conditions[J]. Computers & Mathematics with Applications, 2009, 57(8): 1389-1399.
[51] Q.L.Wang, S.J.Li. Generalized higher-order optimality conditions for set-valued optimization under Henig efficiency[J]. Numerical Functional Analysis and Optimization, 2009, 30(7-8): 849-869.
[52] Q.L.Wang, S.J.Li. Higher-order weakly generalized adjacent epiderivatives and applications to duality of set-valued optimization[J]. Journal of Inequality and Applications, 2009, Art. No462637:1-18.
[53] P.Q.Khanh, N.D.Tuan. Variational sets of multivalued mappings and a unified study of optimality conditions[J]. Journal of Optimization Theory and Applications, 2008, 139:47-65.
[54] P.Q.Khanh, N.D.Tuan. Higher-order variational sets and higher-order optimality conditions for proper efficiency in set-valued nonsmooth vector optimization[J]. Journal of Optimization Theory and Applications, 2008, 139:243-261.
[55]王其林.一类广义凸集值映射优化问题弱有效解的最优性条件[J].四川师范大学学报(自),2007,30(2):556-559.
[56] Q.L.Wang. High-order Fritz John type optimality conditions for Benson proper efficient solutions in set-valued optimization problems[J]. OR Transactions, 2009,13(3):1-9.
[57]刘三阳,盛宝怀.非凸向量集值优化Benson真有效解的最优性条件与对偶[J].应用数学学报,2003,26(2):337-344.
[58] S. J. Li, X. Q. Yang, G. Y. Chen. Nonconvex vector optimization of set-valued mappings[J]. Journal of Mathematical Analysis and Applications, 2003, 283:337-350.
[59] D.T.Luc. On duality theory in multiobjective programming[J]. Journal of Optimization Theory and Applications, 1984, 43(4):557-582.
[60] H.C. Lai, C.P.Ho. Duality theorem of nondifferentiable convex multiobjective programming[J]. Journal of Optimization Theory and Applications, 1986, 50(3):407-420.
[61]应玫茜,择一定理与多目标[J].系统科学与数学,1989,9(1):69-76.
[62] E.V. Tamminen. Sufficient conditions for the existence of multipliers and Lagrangian duality in abstract optimization problems[J]. Journal of Optimization Theory and Applications, 1994, 82(1):93-104.
[63] C.R.Beter, M.Singh. Duality for multiobjective B-invex programming involving n-set function[J]. Journal of Mathematical Analysis and Applications, 1996, 202:701-726.
[64] W. Song. Lagrangian duality for minimization of nonconvex multifunctions[J]. Journal of Optimization Theory and Applications, 1997, 93(1):297-316.
[65] R. Osuna-Gomez, A. Rufian-Lizana, P. Ruiz-Canales. Duality in nondifferetiable vector programming[J]. Journal of Mathematical Analysis and Applications, 2001, 259:462-475.
[66]王其林,李泽民.一类向量极值问题的最优性条件和Lagrange对偶[J].重庆大学学报(自),2005,28(6):106-109.
[67] T. Weir, B. Mond. Generalized convexity and duality in multiple objective programming[J]. Bulletin of the Australian Mathematical Society, 1989, 39:287-299.
[68] P. Q. Khanh. Sufficient optimality conditions and duality in vector optimization with invex-convexlike functions[J]. Journal of Optimization Theory and Applications, 1995, 87: 359-378.
[69] P. H. Sach, B. D. Craven. Invex multifunctions and duality[J]. Numerical Functional Analysis and Optimization, 1991,12:575-591.
[70] P. H. Sach, N. D. Yen, B. D. Craven. Generalized invexity and duality theorems with multifunctions[J]. Numerical Functional Analysis and Optimization, 1994, 15:131-153.
[71] S. K. Mishra, S. Y.Wang, K. K. Lai. Optimality and duality in nondifferentiable and multiobjective programming under generalized d -invexity[J]. Journal of Global Optimization, 2004, 29: 425-438.
[72] V. Preda, K.Koller. General Mond-Weir duality for multiobjectiv programming with generalized ( F ,ρ,θ)-convex set functions[J]. Romanian Journal of Pure and Application Mathematics, 2000, 45:1005-1018.
[73] S.J. Li, K.L. Teo, X. Q. Yang. Higher-order Mond-Weir duality for set-valued optimization[J]. Journal of Computational and Applied Mathematics, 2008, 217(2):339-349.
[74] A.M.Rubinov, X.Q.Yang. Lagrange type Functions in Constrained Nonconvex Optimization[M]. Kluwer Academic Publishers, Dordrecht,2003.
[75] R.T.Rockafellar, R.J.B.Wets. Variational Analysis[M]. Springer-Verlag, Berlin, 1998.
[76] C.J.Goh, X.Q.Yang. Duality in Optimization and Variational Inequalities[M]. Taylor and Francis, London, 2002.
[77] C.J.Goh, X.Q.Yang. Nonlinear Lagrangian theory for nonconvex optimization[J]. Journal of Optimization Theory and Applications, 2001, 109(1):99-121.
[78] C.Y.Wang, X.Q.Yang, X.M.Yang. Unified nonlinear Lagrangian approach to duality and optimal paths[J]. Journal of Optimization Theory and Applications, 2007,135(1):85-100.
[79] X.X.Huang, X.Q.Yang. A unified augmented Lagrangian approach to duality and exact penalization[J]. Mathematics of Operations Research, 2003, 28(3):533-552.
[80] X.Q.Yang, X.X.Huang. A ninlinear Lagrangian approachto constrained optimization problems[J]. SIAM Journal on Optimization, 2001, 11(4):1119-1144.
[81] Y.Y.Zhou, X.Q.Yang. Augmented Lagrangian function, non-quadratic growth condition and exact penalization[J]. Operations Research Letters, 2006, 34:127-134.
[82] Y.Y.Zhou, X.Q.Yang. Some results about duality and exact penalization[J]. Journal of Global Optimization, 2004, 29:497-509.
[83] A.M.Rubinov, X.X.Huang, X.Q.Yang. The zero duality gap property and lower semicontinuity of the perturbation function[J]. Mathematical Methods of Operations Research, 2002, 27(4): 775-791.
[84] W.Fenchel. On conjugate convex functions[J]. Canadian Journal of Mathematics, 1949, 1: 73- 77.
[85] J.J. Moreau. Inf-convolution des fonctions numeriques sur un espace vectoriel[J]. comptes Rendus Academie Sciences Paris, 1963, 256:5047-5049.
[86] R.T.Rockafellar. Convex Analysis[M]. Princeton University Press, Princeton,New Jersey,1970.
[87] R.T.Rockafellar. Conjugate duality and optimization[M]. CBMS-NSF Reqional Conference Series in Applied Mathematics 16, SIAM, Philadelphia, 1974.
[88] I. Ekeland, R. Temam. Convex Analysis and Variational Problems[M]. North-Holland Publishing Company, Amsterdam, Holland, 1976.
[89] T.Tanino, Y.Sawaragi. Conjugate maps and duality in multiobjective optimization[J]. Journal of Optimization Theory and Applications, 1980,31:473-499.
[90]汪寿阳.多目标最优化中的共轭对偶理论[J].系统科学与数学,1984,4(4):303-313.
[91] T.Tanino. Conjugate duality in vector optimization[J]. Journal of Mathematical Analysis and Applications, 1992,167:84-97.
[92] L. Altangerel, R.I.Bot, G. Wanka. Conjugate duality in vector optimization and some applications to the vector variational inequality[J]. Journal of Mathematical Analysis and Applications, 2007, 329:1010-1035.
[93] L. Altangerel, R.I.Bot, G. Wanka. Variational principles for vector equilibrium problems related to conjugate duality[J]. Journal of Nonlinear and Convex Analysis, 2007,8:179-196.
[94] A. Lohne. Optimization with set relations:conjugate duality[J]. Optimization, 2005, 54(3): 265-282.
[95] A. Lohne, C. Tammer. A new approach to duality in vector optimization[J]. Optimization, 2007, 56(1-2): 221-239.
[96] C.S.Lalitha, R.Arora. Conjugate maps, subgradients and conjugate duality in set-valued optimization[J]. Numerical Functional Analysis and Optimization, 2007, 28(7-8):897-909.
[97] F.Giannessi(ed.). Vector Variational Inequalities and Vector Equilibria:Mathematical Theories[C]. Kluwer Academic Publishers, Dordrecht, Holland, 2000.
[98] A.G?pfert, C.Tammer, H.Riahi, C.Zǎlinescu. Variational Methods in Partially Ordered Spaces [M]. Springer, Berlin, 2003.
[99] A.V. Fiacco. Introduction to Sensitivity and Stablity Analysis in Nonlinear Programming[M]. Academic Press, New York, 1983.
[100] R.T. Rockafellar. Lagrange multipliers and subderivatives of optimal value functions in nonlinear programming[J]. Mathematical Programming Study, 1982,17:28-66.
[101] J.P.Aubin, I.Ekeland. Applied Nonlinear Analysis[M]. John Wiley & Sons, New York,1984.
[102] C. Berge. Topological Spaces[M]. Oliver and Boyd, London, 1963.
[103] W.W. Hogan. Point-to-set maps in mathematical programming[J]. SIAM Review, 1973, 15: 591-603.
[104] T.Tanino, Y. Aawaragi. Stability of nondominated solutions in multicriteria decision- marking[J]. Journal of Optimization Theory and Applications, 1980, 30:229-253.
[105] E. Jurkiewicz. Stability of compromise solution in multicriteria decision-making problems [J]. Journal of Optimization Theory and Applications, 1983, 40:77–83.
[106] W.Alt. Local stability of solutions to differentiable optimization problems in Banach spaces, Journal of Optimization Theory and Applications, 1991, 70(3): 443-466.
[107] G. Y. Chen, B. D. Craven. Existence and continuity of solutions for vector optimization[J]. Journal of Optimization Theory and Applications, 1994, 81:459-468.
[108] R. E. Lucchetti, E. Miglierina. Stability for convex vector optimization problems[J]. Optimization, 2004, 53:517-528.
[109] S.J. Li, G.Y. Chen, K.L. Teo. On the stability of generalized vector quasivariational inequality problems[J]. Journal of Optimization Theory and Applications, 2002, 113:283–295.
[110] Y. H. Cheng, D. L. Zhu. Global stability results for the weak vector variational inequality[J]. Journal of Global Optimization, 2005, 32:543–550.
[111] P.Q. Khanh, L.M.Luu. Upper semicontinity of the solution set to parametric vector quasivariational inequalities[J]. Journal of Global Optimization, 2005, 32:569–580.
[112] L. Q. Anh,P. Q. Khanh. On the H?lder continuity of solutions to parametric multivalued vector equilibrium problems[J]. Journal of Mathematical Analysis and Applications, 2006, 321: 308-315.
[113]傅俊义,具移动锥的参数向量均衡问题[J].南昌大学学报,2006, 30:109-111.
[114] S.J.Li, C.R.Chen. Stability of weak vector variational inequality[J]. Nonlinear Analysis:Theory, Methods & Applications, 2009, 70(4):1528-1535.
[115] C.R.Chen, S.J.Li. Semicontinuity of the solution set maps to a set-valued weak vector varivational inequality[J]. Journal of Industrial and Management Optimization, 2007, 3: 519- 528.
[116] S.J.Li, Z.M.Fang. On the stability of a dual weak vector variational inequality problem[J].Journal of Industrial and Management Optimization, 2008,4:155-165.
[117] S.W.Xiang, W.S.Yin, Stability results for efficient solutions of vector optimization problems, Journal of Optimization Theory and Applications, 2007,134:385-398.
[118] Q.L.Wang, S.J.Li. Stability analysis of second-order adjacent derivatives in multiobjective optimization[J]. 2009, submitted.
[119] T.Tanino. Sensitivity analysis in multiobjective optimization[J]. Journal of Optimization Theory and Applications, 1988, 56(3):479-499.
[120] T.Tanino, Stability and sensitivity Analysis in convex vector Optimization[J]. SIAM J. Control Optim, 1988,26:521-536.
[121] D. S. Shi. Contingent derivative of the perturbation map in multiobjective optimization[J]. Journal of Optimization Theory and Applications, 1991, 70:385-396.
[122] D. S. Shi, Sensitivity analysis in convex vector optimization[J]. Journal of Optimization Theory and Applications,1993,77(1):145-159.
[123] H. Kuk, T. Tanino, M. Tanaka. Sensitivity analysis in parametrized convex vector optimization[J]. Journal of Mathematical Analysis and Applications, 1996, 202:511-522.
[124] H. Kuk, T. Tanino, M. Tanaka, Sensitivity analysis in vector optimization[J]. Journal of Optimization Theory and Applications, 1996, 89(3):713-730.
[125] S. J. Li. Sensitivity and stability for contingent derivative in multiobjective optimization[J]. Mathematica Applicata, 1998, 11(2):49-53.
[126] W. Song, L. J. Wan. Contingent epidifferentiability of the value map in vector optimization[J]. Journal of Natural Science of Helongjiang University, 2005, 22(2):198-203.
[127] S. J. Li, H. Yan, G. Y. Chen. Differential and sensitivity properties of gap functions for vector variational inequalities[J]. Mathematical Methods of Operations Research, 2003,57:377-391.
[128] K. W. Meng, S. J. Li. Differential and sensitivity properties of gap functions for Minty vector variational inequalities[J]. Journal of Mathematical Analysis and Applications, 2008, 337: 386- 398.
[129] S. J. Li, K. W. Meng, J.P.Penot. Calculus rules for derivatives of multimaps[J]. Set-Valued and Variarional Analysis, 2009,17:21-39.
[130] M.H.Li, S. J. Li. Second-order differential and sensitivity properties of weak vector variational inequalities[J]. Journal of Optimization Theory and Applications, 2010,144(1):76-87.
[131] Q.L.Wang, S.J.Li. Higher order sensitivity analysis in non-convex vector optimization[J]. Journal of Industrial and Management Optimization, 2010, 6(2):381-392.
[132] C. R. Chen, S. J. Li. Different conjugate dual problems in vector optimization and their relations[J]. Journal of Optimization Theory and Applications, 2009, 140(3):443-461.
[133] C. R. Chen, S. J. Li. Conjugate dual problems in constrained set-valued optimization and applications[J]. European Journal of Operational Research, 2009, 196(1): 21-32.
[134] X.Q.Yang. Directional derivatives for set-valued mappings and applications[J]. Mathematical Methods of Operations Research, 1998, 48:273-285.
[135] J. M.Borwein,D. Zhuang. Supper efficiency in vector optimization[J]. Transactions of the American Mathematical Society, 1993, 338:105-122.
[136] B.Jiménez, V.Novo. Second-order necessary conditions in set constrained differentiable vector optimization[J]. Mathematical Methods of Operations Research, 2003, 58: 299-317.
[137] C.Certh, P.Weidner. Nonconvex separation theorems and some applications in vector optimization[J]. Journal of Optimization Theory and Applications, 1990, 67:297-320.
[138] R.B. Holmes. Geometric Functional Analysis and Its Applications[M]. Springer-Verlag, New York, New York, 1975.
[139] N.J.Huang, J.Li, H.B.Thompson. Stability for parametric implicit vector equilibrium problems[J]. Mathematical and Computer Modelling, 2006,43:1267-1274.
[140] X.H.Gong, J.C.Yao. Lower semicontinuity of the set of efficient solutions for generalized systems[J]. Journal of Optimization Theory and Applications, 2008,138:197-205.