最优化问题的对偶理论与适定性研究
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摘要
对偶理论是最优化问题研究的重要内容，在讨论最优化问题的最优性条件、设计算法等方面有着重要的作用。本文我们主要研究了分式规划问题和向量优化问题的共轭对偶以及极小极大分式规划问题的二阶对偶。同时我们讨论了拟似变分不等式问题的参数适定性。
在第2章中，我们讨论一类目标函数是DC函数（凸函数之差）之比，约束条件是有限多个DC函数的不等式约束的分式规划问题。利用Dinkelbach变换将分式规划问题转化为相应的目标函数是DC函数，约束条件是有限多个DC不等式约束的DC规划问题，然后利用共轭对偶的方法构造了该DC规划问题的Fenchel-Lagrange对偶问题，并且在广义内点的约束规则下证明了弱对偶和强对偶定理。应用对偶定理，我们得到了分式规划问题的Farkas型结论，进一步利用相关共轭函数的上境图给出了Farkas型结论的等价表达。
在第3章中，我们研究向量优化问题的共轭对偶。一方面，利用凸标量优化问题的共轭对偶方法，构造多目标双层优化问题的两种对偶模型，并且基于原问题的真有效解和弱有效解的概念分别得到了原问题与对偶问题之间的弱对偶和强对偶定理。另一方面，我们利用集值映射的ε-共轭映射的概念，给出了集值优化问题的共轭对偶模型，证明了原问题与对偶问题之间的弱对偶和强对偶定理，并且引入原问题的新的Lagrange函数，得到了ε-鞍点存在的充分必要条件。
在第4章中，我们讨论极小极大分式规划问题的二阶对偶。首先引入一类广义二阶凸函数的概念（即二阶(F,α,ρ,d,p)-univex函数）。通过实例说明它是已有文献中的若干二阶凸函数概念的推广。在第2节中，考虑了可微极小极大分式规划问题的两种对偶问题，在相应函数是广义二阶凸的条件下得到了原问题与对偶问题之间的二阶弱对偶、强对偶和严格逆对偶定理。在第3节中，考虑了不可微极小极大分式规划问题的两种对偶问题，得到了相应的二阶对偶定理。
在第5章中，我们在Banach空间中讨论拟似变分不等式问题的参数适定性。引入了拟似变分不等式问题的参数适定性和广义参数适定性的概念，首先建立了拟似变分不等式问题的参数适定性的度量刻画，并且在一定条件下证明了拟似变分不等式问题的参数适定性等价于它的解的存在性和唯一性。然后利用逼近解集的非紧测度给出了拟似变分不等式问题的广义参数适定性的充分必要条件。
Duality theory is one of the most important contents in optimization prob-lems, which plays an important role in discussing the optimality conditions foroptimization problems, algorithm design and so on. In this thesis, our main aimsare to study the conjugate duality for fractional programming and vector optimiza-tion problems and the second order duality for minimax fractional programming.Besides, we also discuss the parametric well-posedness for quasivariational-likeinequality problems.
In Chapter2, we discuss a type of fractional programming problem with aratio of DC functions (diference of convex functions) as objective function andfnitely many inequalities of DC functions as constraints. By using the Dinkelbachtransformation, we convert the fractional programming problem to a DC program-ming problem with the DC objective function and fnitely many DC inequalitiesconstraints. For the DC programming problem, the Fenchel-Lagrange type dualproblem is established, and the weak and strong duality theorems are proved un-der the assumptions of generalized interior point constraint qualifcation. Then wederive the Farkas type results of the fractional programming problem by using theabove duality theorems. Moreover, using the epigraphs of the involved conjugatefunctions, we give an equivalent statement for the Farkas type result.
In Chapter3, we study the conjugate duality for vector optimization prob-lems. On the one hand, by using the conjugate duality approach for convex scalaroptimization problems, we construct two dual models for a multiobjective bilevel optimization problem, obtain the weak and strong duality theorems between theprimal problem and its dual problems based on the properly efcient solutionsand weakly efcient solutions of the primal problem, respectively. On the otherhand, we construct a conjugate dual model for a set-valued optimization problemby using the ε-conjugate map of a set-valued function, and prove some weak andstrong duality theorems between the primal problem and its dual problem. Thenwe introduce a new Lagrange function for the primal problem, and obtain somenecessary and sufcient conditions for the existence of the ε-saddle points.
In Chapter4, we consider the second order duality for minimax fractionalprogramming problems. First, we introduce a new generalized second order convexfunction, which is called second order (F, α, ρ, d, p)-univex function. Then we givesome examples to illustrate that it is an extension of some known second orderconvex function in the literatures. In section2, we consider two dual problems for adiferentiable minimax fractional programming, and establish second order weak,strong and strictly converse duality theorems between the primal problem andits dual problems under the conditions of above generalized second order convexfunctions. In section3, we consider two dual problems for a nondiferentiableminimax fractional programming, and obtain the related second order dualitytheorems.
In Chapter5, we discuss the parametric well-posedness for quasivariational-like inequality problems in Banach spaces. We introduce the notions of the para-metric well-posedness and the parametric well-posedness in generalized sense forquasivariational-like inequality problems. First, we establish some metric char-acterizations of the parametric well-posedness for quasivariational-like inequality problems, and prove that under suitable conditions, the parametric well-posednessis equivalent to the existence and uniqueness of their solutions. Then, we give somenecessary and efcient conditions for the parametric well-posedness in generalizedsense of quasivariational-like inequality problems by using the noncompact mea-sure of the approximating solution sets.
在第2章中，我们讨论一类目标函数是DC函数（凸函数之差）之比，约束条件是有限多个DC函数的不等式约束的分式规划问题。利用Dinkelbach变换将分式规划问题转化为相应的目标函数是DC函数，约束条件是有限多个DC不等式约束的DC规划问题，然后利用共轭对偶的方法构造了该DC规划问题的Fenchel-Lagrange对偶问题，并且在广义内点的约束规则下证明了弱对偶和强对偶定理。应用对偶定理，我们得到了分式规划问题的Farkas型结论，进一步利用相关共轭函数的上境图给出了Farkas型结论的等价表达。
在第3章中，我们研究向量优化问题的共轭对偶。一方面，利用凸标量优化问题的共轭对偶方法，构造多目标双层优化问题的两种对偶模型，并且基于原问题的真有效解和弱有效解的概念分别得到了原问题与对偶问题之间的弱对偶和强对偶定理。另一方面，我们利用集值映射的ε-共轭映射的概念，给出了集值优化问题的共轭对偶模型，证明了原问题与对偶问题之间的弱对偶和强对偶定理，并且引入原问题的新的Lagrange函数，得到了ε-鞍点存在的充分必要条件。
在第4章中，我们讨论极小极大分式规划问题的二阶对偶。首先引入一类广义二阶凸函数的概念（即二阶(F,α,ρ,d,p)-univex函数）。通过实例说明它是已有文献中的若干二阶凸函数概念的推广。在第2节中，考虑了可微极小极大分式规划问题的两种对偶问题，在相应函数是广义二阶凸的条件下得到了原问题与对偶问题之间的二阶弱对偶、强对偶和严格逆对偶定理。在第3节中，考虑了不可微极小极大分式规划问题的两种对偶问题，得到了相应的二阶对偶定理。
在第5章中，我们在Banach空间中讨论拟似变分不等式问题的参数适定性。引入了拟似变分不等式问题的参数适定性和广义参数适定性的概念，首先建立了拟似变分不等式问题的参数适定性的度量刻画，并且在一定条件下证明了拟似变分不等式问题的参数适定性等价于它的解的存在性和唯一性。然后利用逼近解集的非紧测度给出了拟似变分不等式问题的广义参数适定性的充分必要条件。
Duality theory is one of the most important contents in optimization prob-lems, which plays an important role in discussing the optimality conditions foroptimization problems, algorithm design and so on. In this thesis, our main aimsare to study the conjugate duality for fractional programming and vector optimiza-tion problems and the second order duality for minimax fractional programming.Besides, we also discuss the parametric well-posedness for quasivariational-likeinequality problems.
In Chapter2, we discuss a type of fractional programming problem with aratio of DC functions (diference of convex functions) as objective function andfnitely many inequalities of DC functions as constraints. By using the Dinkelbachtransformation, we convert the fractional programming problem to a DC program-ming problem with the DC objective function and fnitely many DC inequalitiesconstraints. For the DC programming problem, the Fenchel-Lagrange type dualproblem is established, and the weak and strong duality theorems are proved un-der the assumptions of generalized interior point constraint qualifcation. Then wederive the Farkas type results of the fractional programming problem by using theabove duality theorems. Moreover, using the epigraphs of the involved conjugatefunctions, we give an equivalent statement for the Farkas type result.
In Chapter3, we study the conjugate duality for vector optimization prob-lems. On the one hand, by using the conjugate duality approach for convex scalaroptimization problems, we construct two dual models for a multiobjective bilevel optimization problem, obtain the weak and strong duality theorems between theprimal problem and its dual problems based on the properly efcient solutionsand weakly efcient solutions of the primal problem, respectively. On the otherhand, we construct a conjugate dual model for a set-valued optimization problemby using the ε-conjugate map of a set-valued function, and prove some weak andstrong duality theorems between the primal problem and its dual problem. Thenwe introduce a new Lagrange function for the primal problem, and obtain somenecessary and sufcient conditions for the existence of the ε-saddle points.
In Chapter4, we consider the second order duality for minimax fractionalprogramming problems. First, we introduce a new generalized second order convexfunction, which is called second order (F, α, ρ, d, p)-univex function. Then we givesome examples to illustrate that it is an extension of some known second orderconvex function in the literatures. In section2, we consider two dual problems for adiferentiable minimax fractional programming, and establish second order weak,strong and strictly converse duality theorems between the primal problem andits dual problems under the conditions of above generalized second order convexfunctions. In section3, we consider two dual problems for a nondiferentiableminimax fractional programming, and obtain the related second order dualitytheorems.
In Chapter5, we discuss the parametric well-posedness for quasivariational-like inequality problems in Banach spaces. We introduce the notions of the para-metric well-posedness and the parametric well-posedness in generalized sense forquasivariational-like inequality problems. First, we establish some metric char-acterizations of the parametric well-posedness for quasivariational-like inequality problems, and prove that under suitable conditions, the parametric well-posednessis equivalent to the existence and uniqueness of their solutions. Then, we give somenecessary and efcient conditions for the parametric well-posedness in generalizedsense of quasivariational-like inequality problems by using the noncompact mea-sure of the approximating solution sets.
引文
1W. Fenchel. On conjugate convex functions, Canadian J. Math.,1949,1:73-77.
2J. J. Moreau. Inf-convolutiuon des fonctions mumeriques sur un espacevectoriel. Comptes Rendus Academie Sciences Pairs,1963,256:5047-5049.
3R. T. Rockafellar. Convex analysis. Princeton University Press, Princeton,1970.
4I. Ekeland, R. Temam. Convex analysis and variational problems. North-Holland Publishing Company, Amsterdam,1976.
5C. Zalinescu. Convex analysis in general vector spaces. World Scientifc,2002.
6G. Wanka, R. I. Bo t. On the relations between diferent dual problems inconvex mathematical programming. In: Operational Research Proceedings2001, P. Chamoni, R. Leisten, A. Martin, J. Minnemann and H. Stadtler(Eds.), Springer Verlag, Berlin,2002,255-262.
7R. I. Bo t, G. Kassay，G. Wanka. Strong duality for generalized convexoptimization problems. J. Optim. Theor. Appl.,2005,127(1):45-70.
8R. I. Bo t, G. Kassay，G. Wanka. Duality for almost convex optimizationproblems via the perturbation approach. J. Glob. Optim.,2008,42:385-399.
9R. I. Bo t, G. Wanka. Farkas-results with conjugate functions. SIAM J.Optim.,2005,15:540-554.
10R. I. Bo t, I. B. Hodrea, G. Wanka. Some new Farkas-type results for in-equality system with DC functions. J. Glob. Optim.,2007,39:595-608.
11R. I. Bo t, I. B. Hodrea, G. Wanka. Farkas-type results for inequality systemswith composed convex functions via conjugate duality. J. Math. Anal.Appl.,2006,322(1):316-328.
12R. I. Bo t, I. B. Hodrea, G. Wanka. Farkas-type results for fractional pro-gramming problems. Nonlinear Anal-Theor.,2007,67:1690-1703.
13X. H. Zhang, C. Z. Cheng. Some Farkas-type results for fractional program-ming with DC functions. Nonlinear Anal-Real.,2009,10:1679-1690.
14V. Jeyakumar. Farkas’ Lemma: Generalizations. In: Encyclopedia of Op-timization, vol. II, C.A.Floudas and P. Pardalos (Eds.), Kluwer AcademicPublishers, Dordrecht, The Netherlands,2001,87-91.
15V. Jeyakumar. Characterizing set containments involving infnite convexconstraints and reverse-convex constraints. SIAM J. Optim.,2003,13:947-959.
16V. Jeyakumar, G. M. Lee. Complete characterization of stable Farkas’lemma and coneconvex programming duality. Math. Program. Ser. A,2008,114:335–347.
17W. Li, C. Nahak, I. Singer. Constraint qualifcations for semi-infnite systemsof convex inequalities. SIAM J. Optim.,2000,11:31-52
18N. Dinh, M. A. Goberna, M. A. L′opez. From linear to convex systems:consistency, Farkas lemma and applications. J. Convex Anal.,2006,13:279–290.
19N. Dinh, M. A. Goberna, M. A. Lo′pez, T. Q. Son. New Farkas-type con-straint qualifcations in convex infnite programming. ESAIM Control Op-tim. Calc. Var.,2007,13:580–597.
20N. Dinh, G. Vallet, T. T. A. Nghia. Farkas-type results and duality for DCprograms with convex constraints. J. Convex Anal.,2008,15:235–262.
21D. H. Fang, C. Li, K. F. NG. Constraint qualifcations for extended farkas’slemmas and lagrangian dualities in convex infnite programming. SIAM J.Optim.,2009,20(3):1311-1332.
22A. Aboussoror, S. Adly. A Fenchel-Lagrange duality approach for a bilevelprogramming problem with extremal-value function. J. Optim. TheoryAppl.,2011,149:254-268.
23R. S. Burachik, V. Jeyakumar. A dual condition for the convex subdifer-ential sum formula with applications. J. Conv. Anal.,2005,12(2):279-290.
24R. S. Burachik, V. Jeyakumar. A simpler condition for the normal coneintersection formula. Proc. Amer. Math. Soc.,2005,133(6):1741-1748.
25R. I. Bo t, G. Wanka. A weaker regularity condition for subdiferential cal-culus and Fenchel duality in infnite dimensional spaces. Nonlinear Anal-Theor.,2006,64:2787-2804.
26R. I. Bo t, S. M. Grad, G. Wanka. A new constraint qualifcation for theformula of the subdiferential of composed convex functions in infnite di-mensional spaces. Math. Nachr.,2008,281:1088-1107.
27R. I. Bo t, I. B. Hodrea, G. Wanka. ε-Optimality conditions for composedconvex optimization problems. J. Approx. Theory,2008,153:108-121.
28R. I. Bo t, S. M. Grad, G. Wanka. New regularity conditions for Lagrangeand Fenchel-Lagrange duality in infnite dimensional spaces. Math. Ineq.Appl.,2009,12(1):171-189.
29A. Hantoute, M. A. Lo′pez, C. ZA linescu. Subdiferential calculus rules inconvex analysis: A unifying approach via pointwise supremum functions.SIAM J. Optim.,2008,19:863-882.
30R. I. Bo t, S. M. Grad, G. Wanka. Generalized Moreau-Rockafellar resultsfor composed convex functions. Optimization,2009,58(7):917-933.
31C. Li, K. F. Ng, On constraint qualifcation for infnite system of convexinequalities in a Banach space. SIAM J. Optim.,2005,15:488-512.
32C. Li, K. F. Ng, T.K. Pong. Constraint qualifcations for convex inequalitysystems with applications in constrained optimization. SIAM J. Optim.,2008,19:163-187.
33R. I. Bo t. Conjugate duality in convex optimization. Lecture Notes inEconomics and Mathematical Systems, Springer-Verlag Berlin Heidelberg,2010.
34L. Altangerel, R. I. Bo t, G. Wanka. On gap functions for equilibrium prob-lems via Fenchel duality. Pacifc J. Optim.,2006,2(3):667-678.
35L. Altangerel, R. I. Bo t, G. Wanka. On the construction of gap functionsfor variational inequalities via conjugate duality. Asia-Pacifc J. Oper.Res.,2007,4(3):353-371.
36R. I. Bo t, A. E. Capata. Existence results and gap functions for the general-ized equilibrium problem with composed functions. Nonlinear Anal-Theor.,2010,72:316-324.
37N. Dinh, J. J. Strodiot, V. H. Nguyen. Duality and optimality conditions forgeneralized equilibrium problems involving DC functions. J. Glob. Optim.,2010,48:183-208.
38G. Bigi, M. Castellani, G. Kassay. A Dual view of equilibrium problems. J.Math. Aanl. Appl.,2008,324:17-26.
39J. E. Mart′nez-Legaz, W. Sosa. Duality for equilibrium problems. J. Glob.Optim.,2006,35:311-319.
40M. Soleimani-damaneh. The gap function for optimization problems in Ba-nach spaces. Nonlinear Anal-Theor,2008,69:716-723.
41X. K. Sun, S. J. Li. Duality and Farkas-type results for extended Ky Faninequalities with DC functions. Optim. Lett., DOI:10.1007/s11590-011-0434-8.
42L. Altangerel. A duality approach to gap functions for variational inequal-ities and equilibrium problems, PhD Dissertation, Faculty of Mathematics.Chemnitz University of Technology,2006.
43D. Gale, H. W. Kuhn, A. W. Tucker. Linear programming and the theory ofgames. In: Activity Analysis of Production and Allocation, T. C. Koopmans(Ed.) John Wiley&Sons, New York,1951,317-329.
44W. Ro¨dder. A generalized saddle point theory: its application to dualitytheory for linear vector optimum problems. Eur. J. Oper. Res.,1977,1(1):55-59.
45H. Isermann. On some relations between a dual pair of multiple objectivelinear programs. Zeitschrift fur Oper. Res.,1978,22:33-41.
46T. Tanino, Y. Sawargi. Conjugate maps and duality in multiobjective opti-mization. J. optim. Theory Appl.,1980,31:473-499.
47D. T. Luc. Theory of vector optimization. Springer, Berlin,1989.
48H.Kawasaki. A duality theorem in multiobjective nonlinear programming.Math. Oper. Res.,1982,7:95-110.
49T. Tanino. Conjugate duality in vector optimization. J. Math. Anal. Appl.,1992,167:84-97.
50W. Song. Conjugate duality in set-valued vector optimization. J. Math.Anal. Appl.,1997,216:265-283.
51W. Song. A generalization of Fenchel duality in set-valued vector optimiza-tion. Math. Meth. Oper. Res.,1998,48:259-272.
52C. R. Chen, S. J. Li. Diferent conjugate dual problems in vector optimiza-tion and their relations. J. Optim. Theory Appl.,2009,140:443–461.
53S. J. Li, C. R. Chen, S. Y. Wu. Conjugate dual problems in constrainedset-valued optimization and applications. Europ. J. Oper. Res.,2009,196:21-32.
54S. J. Li, X. K. Sun, H. M. Liu, S. F. Yao, K. L. Teo. Conjugate duality inconstrained set-valued vector optimization problems. Numer. Funct. Anal.Optim.,2011,32(1):65–82.
55J. H. Jia, Z. F. Li. ε-Conjugate maps and ε-conjugate duality in vectoroptimization with set valued maps. Optimization,2008,57(5):621-633.
56L. Altangerel, R. I. Bo t, G. Wanka. Conjugate duality in vector optimizationand some applications to the vector variational inequality. J. Math. Anal.Appl.,2007,329:1010-1035.
57L. Altangerel, R. I. Bo t, G. Wanka. Variational principles for vector equi-librium problems related to conjugate duality. J. Nonlinear Convex Anal.,2007,8:179-196.
58G. Y. Chen, C. J. Goh, X. Q. Yang. On gap functions for vector variationalinequalities. In: Vector Variational Inequalities and Vector Equilibria, F.Giannessi (ed.), Kluwer Academic Publishers, Dordrecht, Holland,2000,55-72.
59S. J. Li. On the ε-variational principle for set-valued mappings. Comput.Math. Appl.,2007,53:1627-1632.
60S. J. Li, S. F. Yao, C. R. Chen. Saddle points and gap functions for vectorequilibrium problems via conjugate duality in vector optimization. J. Math.Anal. Appl.,2008,343:853-865.
61S. J. Li, P. Zhao. A method of duality for a mixed vector equilibriumproblem. Optim. Lett.,2010,4:85-96.
62P. H. Sach. New generalized convexity notion for set-valued maps and appli-cation to vector optimization. J. Optim. Theory Appl.,2005,125:157-179.
63P. H. Sach. Moreau-Rockafellar Theorems for Nonconvex Set-Valued Maps.J. Optim. Theory Appl.,2007,133:213-227.
64P. Wolfe. A duality theorem for nonlinear programming. Quart. Appl.Math.,1961,19:239-244.
65B. Mond, T. Weir. Generalized concavity and duality. In: Generalizedconcavity in Optimization and Economics, S. Schaible and W.T. Ziemba,(Eds.), Academic Press, New York,1981,263-279.
66B. Mond, R. R. Egudo. On strict converse duality in nonlinear programming.J. Infor. optim. Sci.,1985,6:113-116.
67B. Mond, T. Weir, B. D. Craven. On duality in generalized minimax pro-grams. Asia-Pacifc J. Oper. Res.,1990,7:55-62.
68Z. A. Liang, H. X. Huang, P. M. Pardalos. Optimality conditions and dualityfor a class of nonlinear fractional programming problems. J. Optim. TheoryAppl.,2001,110:611-619.
69M. A. Hanson, R. Pini, C. Singh. Multiobjective programming under gen-eralized type-I invexity. J. Math. Anal. Appl.,2001,261:562-577.
70T. Antczak, Multiobjective programming under d-invexity. Eur. J. Oper.Res.,2002,137:28-36.
71Y. L. Ye. d-invexity and optimality conditions. J. Math. Anal. Appl.,1991,153:242-249.
72J. C. Liu, C. S. Wu. On minimax fractional optimality conditions withinvexity. J. Math. Anal. Appl.,1998,219:21-35.
73J. C. Liu, C. S. Wu. On minimax fractional optimality conditions with(F, ρ)-convexity. J. Math. Anal. Appl.,1998,219:36-51.
74S. K. Mishra, S. Y. Wang, K. K. Lai. Nondiferentiable multiobjective pro-gramming under generalized d-univexity. Eur. J. Oper. Res.,2005,160:218–226.
75C. Nahak, S. Nanda. Multiobjective duality with ρ (η, θ) invexity. J.Appl. Math. Stoch. Anal.,2005,2:175-180.
76R. P. Pant, J. S. Rautela. α-invexity and duality in mathematical program-ming. J. Math. Anal. Appr. Theory,2006,1(2):104-114.
77S. K. Mishra, R. P. Pant, J. S. Rautela. Generalized α-invexity and non-diferentiable minimax fractional programming. J. Comput. Appl. Math.,2007,206:122-135.
78S. K. Mishra, R. P. Pant, J. S. Rautela. Generalized α-univexity and dual-ity for nondiferentiable minimax fractional programming. Nonlinear Anal-Theor.,2009,70:144-158.
79O. L. Mangasarian. Second and higher-order duality in nonlinear program-ming. J. Math. Anal. Appl.,1975,51:607-620.
80B. Mond. Second order duality for nonlinear programs. Opsearch.,1974,11:90-99.
81C. R. Bector, B. K. Bector.(Generalized)-bonvex functions and secondorder duality for a nonlinear programming problem. Congr. Number.,1986,52:37-52.
82C. R. Bector, S. Chandra. Second order symmetric and self-dual programs.Opsearch.,1986,23:89-95.
83C. R. Bector, B. K. Bector. On various duality theorems for second orderduality in nonlinear programming. Cahiers Center Etudes Rech. Opre.,1986,28:283-792.
84C. R. Bector, S. Chandra. First and second order duality for a class ofnondiferentiable fractional programming problem. J. Infor. Optim. Sci.,1986,7:335-348.
85C. R. Bector, S. Chandra. Generalized-convexity and higher order dualityfor fractional programming. Opsearch.,1987,24:143-154.
86C. R. Bector, S. Chandra, I. Husain. Second order duality in continuoustime nonlinear programming. Congr. Number.,1987,56:121-134.
87C. R. Bector, S. Chandra, I. Husain. Second order duality for a minimaxprogramming problem. Opsearch.,1991,28:249-263.
88I. Ahmad, Z. Husain. Second order (F, α, ρ, d)-convexity and duality inmultiobjective programming. Inform. Sci.,2006,176:3094–3103.
89H.C. Lai, J.C. Liu, K. Tanaka. Necessary and sufcient conditions for min-imax fractional programming. J. Math. Anal. Appl.,1999,230:311–328.
90I. Ahmad, Z. Husain. Duality in nondiferentiable minimax fractional pro-gramming with generalized convexity. Appl. Math. Comput.,2006,176:545–551.
91Z. Husain, A. Jayswal, I. Ahmad. Second order duality for nondiferentiableminimax programming problems with generalized convexity. J. Glob. Op-tim.,2009,44:593-608.
92I. Ahmad, Z. Husain, S. AI-Homidan. Second-order duality in nondiferen-tiable fractional programming. Nonlinear Anal-Real,2011,12:1103-1110.
93A. N. Tykhonov. On the stability of the functional optimization problem.USSR Comput. Math. Math. Phys.,1966,6:28-33.
94E. S. Levitin, B. T. Polyak. Convergence of minimizing sequences in condi-tional extremum problems. Soviet Mathematics Doklady,1966,7:764-767.
95X. X. Huang, X. Q. Yang. Generalized Levitin-Polyak well-posedness inconstrained optimization. SIAM J. Optim.,2006,17(1):243-258.
96X. X. Huang, X. Q. Yang, D. L. Zhu. Levitin-Polyak well-posedness ofvariational inequality problems with functional constraints. J. Glob. Optim.,2009,44:159-174.
97B. Jiang, J. Zhang, X. X. Huang. Levitin-Polyak well-posedness of gener-alized quasivariational inequalities with functional constraints. NonlinearAnal-Theor,2009,70:1492-1503.
98T. Zolezzi. Well-posedness criteria in optimization with application to thecalculus of variations. Nonlinear Anal.,1995,25:437-453.
99T. Zolezzi. Extended well-posedness of optimization problems. J. Optim.Theory Appl.,1996,91(1):257-266.
100B. Lemaire, C. Ould Ahmed Salem, J.P. Revalski. Well-posedness by per-turbations of variational problems. J, Optim. Theory Appl.,2002,115:345-368.
101Y. P. Fang, N. J. Huang, J. C. Yao. Well-posedness of mixed variationalinequalities, inclusion problems and fxed point problems. J. Glob. Optim.,2008,41:117-133.
102D. Dentcheva, S. Helbig. On variational principles, level sets, well-posednessand ε-solutions in vector optimization. J. Optim. Theory Appl.,1996,89:329-349.
103X. X. Huang. Extended well-posedness properties of vector optimizationproblems. J. Optim. Theory Appl.,2000,106:165-182.
104X. X. Huang. Pointwise well-posedness of perturbed vector optimizationproblems in a vector-valued variational principle. J. Optim. Theory Appl.,2001,108:671-686.
105X. X. Huang, X. Q. Yang. Levitin-Polyak well-posedness of constrainedvector optimization problems. J. Glob. Optim.,2007,37:287-304.
106X. X. Huang, X. Q. Yang. Further study on the Levitin-Polyak well-posedness of constrained convex vector optimization problems. NonlinearAnal.,2012,75:1341-1347.
107G. P. Crespi, A. Guerraggio, M. Rocca. Well posedness in vector optimiza-tion problems and vector variational inequalities. J. Optim. Theory Appl.,2007,132:213-226.
108G. P. Crespi, M. Papalia, M. Rocca. Extended well-posedness of quasiconvexvector optimization problems. J. Optim. Theory Appl.,2009,141:285-297.
109M. Bianchi, G. Kassay, R. Pini. Well-posedness for vector equilibrium prob-lems. Math. Methods Oper. Res.,2009,70:171-182.
110J. W. Peng, S. Y. Wu. The generalized Tykhonov well-posedness for systemof vector quasi-equilibrium problems. Optim. Lett.,2010,4:501-512.
111X. J. Long, N. J. Huang. Metric characterizations of α-well-posedness forsymmetric quasi-equilibrium problems. J. Glob. Optim.,2009,45:459-471.
112S.J. Li, M. H. Li. Levitin-Polyak well-posedness of vector equilibrium prob-lems. Math. Methods Oper. Res.,2009,69:125-140.
113S. J. Li, W. Y. Zhang. Hadamard well-posed vector optimization problems.J. Glob. Optim.,2010,46:383-393.
114C. R. Bector, Programming Problems with Convex Fractional Functions,Indian Institute of Technology,1967,383-391.
115B.D. Craven, B. Mond. The dual of a fractional linear program. J. Math.Anal. Appl.,1973,42:507-512.
116S. Schaible. Fractional programming: applications and algorithms. Eur. J.Oper. Res.,1981,7:111-120.
117S. Chandra, V. Kumar. Duality in fractional minimax programming. J.Austral. Math. Soc. Ser. A,1995,58:376-386.
118X. M. Yang, S. H. Hou. On minimax fractional optimality and duality withgeneralized convexity, J. Glob. Optim.,2005,31(2):235-252.
119W. Dinkelbach. On nonlinear programming. Management Science,1967,13(7):526-543.
120S. S. Kutateladze. Changes of variables in the Young transformation. Sov.Math. Dokl.,1977,18(2):545-548.
121C. J. Goh, X. Q. Yang. Duality in optimization and variational inequalities.Taylor and Francis, London,2002.
122R. I. Bo t, S. M. Grad, G. Wanka. Duality in vector optimization. Springer-Verlag Berlin Heidelberg,2009.
123J. F. Bard. Some properties of the bilevel programming problem. J. Optim.Theory Appl.,1991,68:371-378.
124S. Dempe. A necessary and sufcient optimality condition for bilevel pro-gramming problem. Optimization,1992,25:341-354.
125S. Dempe. First order necessary optimality conditions for general bilevelprogramming problems. J. Optim. Theory Appl.,1997,95:735-739.
126J. J. Ye. Nondiferentiable multiplier rules for optimization and bilevel op-timization problems. SIAM J. Optim.,2004,15:252-274.
127J. J. Ye. Necessary optimality conditions for multiobjective bilevel programs.Math. Oper. Res.,2011,36(1):165-184.
128W. D. Rong, Y. N. Wu. ε-weak minimal solutions of vector optimizationproblems with set-valued maps. J. Optim. Theory Appl.,2000,106:569-579.
129D. Z. Du, P. M. Pardalos. Minimax and applications. Kluwer AcademicPublishers, The Netherlands,1995.
130P. Pandian, G. Natarajan. Second order (b; F)-convexity in multiobjectivenonlinear programming. Int. J. Math. Anal.,2010,4(7):303-314.
131J. Zhang, B. Mond. Second order duality for multiobjective nonlinear pro-gramming involving generalized convexity. In: Proceedings of the Optimiza-tion Miniconference III, B. M. Glover, B. D. Craven and D. Ralph (Eds.),University of Ballarat,1997,79-95.
132S. Chandra, V. Kumar. Duality in fractional minimax programming. J.Austral. Math. Soc. Ser. A,1995,58:376-386.
133M. B. Lignola, J. Morgan. Well-posedness for optimization problems withconstraints defned by variational inequalities having a unique solution. J.Global Optim.,2000,16(1):57-67.
134Y. P. Fang, R. Hu. Parametric well-posedness for variational inequalitiesdefned by bifunctions. Comput. Math. Appl.,2007,53:1306-1316.
135M. B. Lignola. J. Morgan, Semicontinuity and episemicontinuity: equiva-lence and applications. Boll. Unione Mat. Ital.1994,8B:1-6.
136J. Banas, K. Goebel. Measures of noncompactness in Banach spaces. MarcelDekker, New York and Basel,1980.
137J. P. Aubin. Applied nonlinear analysis. New York: Wiley,1984.
2J. J. Moreau. Inf-convolutiuon des fonctions mumeriques sur un espacevectoriel. Comptes Rendus Academie Sciences Pairs,1963,256:5047-5049.
3R. T. Rockafellar. Convex analysis. Princeton University Press, Princeton,1970.
4I. Ekeland, R. Temam. Convex analysis and variational problems. North-Holland Publishing Company, Amsterdam,1976.
5C. Zalinescu. Convex analysis in general vector spaces. World Scientifc,2002.
6G. Wanka, R. I. Bo t. On the relations between diferent dual problems inconvex mathematical programming. In: Operational Research Proceedings2001, P. Chamoni, R. Leisten, A. Martin, J. Minnemann and H. Stadtler(Eds.), Springer Verlag, Berlin,2002,255-262.
7R. I. Bo t, G. Kassay，G. Wanka. Strong duality for generalized convexoptimization problems. J. Optim. Theor. Appl.,2005,127(1):45-70.
8R. I. Bo t, G. Kassay，G. Wanka. Duality for almost convex optimizationproblems via the perturbation approach. J. Glob. Optim.,2008,42:385-399.
9R. I. Bo t, G. Wanka. Farkas-results with conjugate functions. SIAM J.Optim.,2005,15:540-554.
10R. I. Bo t, I. B. Hodrea, G. Wanka. Some new Farkas-type results for in-equality system with DC functions. J. Glob. Optim.,2007,39:595-608.
11R. I. Bo t, I. B. Hodrea, G. Wanka. Farkas-type results for inequality systemswith composed convex functions via conjugate duality. J. Math. Anal.Appl.,2006,322(1):316-328.
12R. I. Bo t, I. B. Hodrea, G. Wanka. Farkas-type results for fractional pro-gramming problems. Nonlinear Anal-Theor.,2007,67:1690-1703.
13X. H. Zhang, C. Z. Cheng. Some Farkas-type results for fractional program-ming with DC functions. Nonlinear Anal-Real.,2009,10:1679-1690.
14V. Jeyakumar. Farkas’ Lemma: Generalizations. In: Encyclopedia of Op-timization, vol. II, C.A.Floudas and P. Pardalos (Eds.), Kluwer AcademicPublishers, Dordrecht, The Netherlands,2001,87-91.
15V. Jeyakumar. Characterizing set containments involving infnite convexconstraints and reverse-convex constraints. SIAM J. Optim.,2003,13:947-959.
16V. Jeyakumar, G. M. Lee. Complete characterization of stable Farkas’lemma and coneconvex programming duality. Math. Program. Ser. A,2008,114:335–347.
17W. Li, C. Nahak, I. Singer. Constraint qualifcations for semi-infnite systemsof convex inequalities. SIAM J. Optim.,2000,11:31-52
18N. Dinh, M. A. Goberna, M. A. L′opez. From linear to convex systems:consistency, Farkas lemma and applications. J. Convex Anal.,2006,13:279–290.
19N. Dinh, M. A. Goberna, M. A. Lo′pez, T. Q. Son. New Farkas-type con-straint qualifcations in convex infnite programming. ESAIM Control Op-tim. Calc. Var.,2007,13:580–597.
20N. Dinh, G. Vallet, T. T. A. Nghia. Farkas-type results and duality for DCprograms with convex constraints. J. Convex Anal.,2008,15:235–262.
21D. H. Fang, C. Li, K. F. NG. Constraint qualifcations for extended farkas’slemmas and lagrangian dualities in convex infnite programming. SIAM J.Optim.,2009,20(3):1311-1332.
22A. Aboussoror, S. Adly. A Fenchel-Lagrange duality approach for a bilevelprogramming problem with extremal-value function. J. Optim. TheoryAppl.,2011,149:254-268.
23R. S. Burachik, V. Jeyakumar. A dual condition for the convex subdifer-ential sum formula with applications. J. Conv. Anal.,2005,12(2):279-290.
24R. S. Burachik, V. Jeyakumar. A simpler condition for the normal coneintersection formula. Proc. Amer. Math. Soc.,2005,133(6):1741-1748.
25R. I. Bo t, G. Wanka. A weaker regularity condition for subdiferential cal-culus and Fenchel duality in infnite dimensional spaces. Nonlinear Anal-Theor.,2006,64:2787-2804.
26R. I. Bo t, S. M. Grad, G. Wanka. A new constraint qualifcation for theformula of the subdiferential of composed convex functions in infnite di-mensional spaces. Math. Nachr.,2008,281:1088-1107.
27R. I. Bo t, I. B. Hodrea, G. Wanka. ε-Optimality conditions for composedconvex optimization problems. J. Approx. Theory,2008,153:108-121.
28R. I. Bo t, S. M. Grad, G. Wanka. New regularity conditions for Lagrangeand Fenchel-Lagrange duality in infnite dimensional spaces. Math. Ineq.Appl.,2009,12(1):171-189.
29A. Hantoute, M. A. Lo′pez, C. ZA linescu. Subdiferential calculus rules inconvex analysis: A unifying approach via pointwise supremum functions.SIAM J. Optim.,2008,19:863-882.
30R. I. Bo t, S. M. Grad, G. Wanka. Generalized Moreau-Rockafellar resultsfor composed convex functions. Optimization,2009,58(7):917-933.
31C. Li, K. F. Ng, On constraint qualifcation for infnite system of convexinequalities in a Banach space. SIAM J. Optim.,2005,15:488-512.
32C. Li, K. F. Ng, T.K. Pong. Constraint qualifcations for convex inequalitysystems with applications in constrained optimization. SIAM J. Optim.,2008,19:163-187.
33R. I. Bo t. Conjugate duality in convex optimization. Lecture Notes inEconomics and Mathematical Systems, Springer-Verlag Berlin Heidelberg,2010.
34L. Altangerel, R. I. Bo t, G. Wanka. On gap functions for equilibrium prob-lems via Fenchel duality. Pacifc J. Optim.,2006,2(3):667-678.
35L. Altangerel, R. I. Bo t, G. Wanka. On the construction of gap functionsfor variational inequalities via conjugate duality. Asia-Pacifc J. Oper.Res.,2007,4(3):353-371.
36R. I. Bo t, A. E. Capata. Existence results and gap functions for the general-ized equilibrium problem with composed functions. Nonlinear Anal-Theor.,2010,72:316-324.
37N. Dinh, J. J. Strodiot, V. H. Nguyen. Duality and optimality conditions forgeneralized equilibrium problems involving DC functions. J. Glob. Optim.,2010,48:183-208.
38G. Bigi, M. Castellani, G. Kassay. A Dual view of equilibrium problems. J.Math. Aanl. Appl.,2008,324:17-26.
39J. E. Mart′nez-Legaz, W. Sosa. Duality for equilibrium problems. J. Glob.Optim.,2006,35:311-319.
40M. Soleimani-damaneh. The gap function for optimization problems in Ba-nach spaces. Nonlinear Anal-Theor,2008,69:716-723.
41X. K. Sun, S. J. Li. Duality and Farkas-type results for extended Ky Faninequalities with DC functions. Optim. Lett., DOI:10.1007/s11590-011-0434-8.
42L. Altangerel. A duality approach to gap functions for variational inequal-ities and equilibrium problems, PhD Dissertation, Faculty of Mathematics.Chemnitz University of Technology,2006.
43D. Gale, H. W. Kuhn, A. W. Tucker. Linear programming and the theory ofgames. In: Activity Analysis of Production and Allocation, T. C. Koopmans(Ed.) John Wiley&Sons, New York,1951,317-329.
44W. Ro¨dder. A generalized saddle point theory: its application to dualitytheory for linear vector optimum problems. Eur. J. Oper. Res.,1977,1(1):55-59.
45H. Isermann. On some relations between a dual pair of multiple objectivelinear programs. Zeitschrift fur Oper. Res.,1978,22:33-41.
46T. Tanino, Y. Sawargi. Conjugate maps and duality in multiobjective opti-mization. J. optim. Theory Appl.,1980,31:473-499.
47D. T. Luc. Theory of vector optimization. Springer, Berlin,1989.
48H.Kawasaki. A duality theorem in multiobjective nonlinear programming.Math. Oper. Res.,1982,7:95-110.
49T. Tanino. Conjugate duality in vector optimization. J. Math. Anal. Appl.,1992,167:84-97.
50W. Song. Conjugate duality in set-valued vector optimization. J. Math.Anal. Appl.,1997,216:265-283.
51W. Song. A generalization of Fenchel duality in set-valued vector optimiza-tion. Math. Meth. Oper. Res.,1998,48:259-272.
52C. R. Chen, S. J. Li. Diferent conjugate dual problems in vector optimiza-tion and their relations. J. Optim. Theory Appl.,2009,140:443–461.
53S. J. Li, C. R. Chen, S. Y. Wu. Conjugate dual problems in constrainedset-valued optimization and applications. Europ. J. Oper. Res.,2009,196:21-32.
54S. J. Li, X. K. Sun, H. M. Liu, S. F. Yao, K. L. Teo. Conjugate duality inconstrained set-valued vector optimization problems. Numer. Funct. Anal.Optim.,2011,32(1):65–82.
55J. H. Jia, Z. F. Li. ε-Conjugate maps and ε-conjugate duality in vectoroptimization with set valued maps. Optimization,2008,57(5):621-633.
56L. Altangerel, R. I. Bo t, G. Wanka. Conjugate duality in vector optimizationand some applications to the vector variational inequality. J. Math. Anal.Appl.,2007,329:1010-1035.
57L. Altangerel, R. I. Bo t, G. Wanka. Variational principles for vector equi-librium problems related to conjugate duality. J. Nonlinear Convex Anal.,2007,8:179-196.
58G. Y. Chen, C. J. Goh, X. Q. Yang. On gap functions for vector variationalinequalities. In: Vector Variational Inequalities and Vector Equilibria, F.Giannessi (ed.), Kluwer Academic Publishers, Dordrecht, Holland,2000,55-72.
59S. J. Li. On the ε-variational principle for set-valued mappings. Comput.Math. Appl.,2007,53:1627-1632.
60S. J. Li, S. F. Yao, C. R. Chen. Saddle points and gap functions for vectorequilibrium problems via conjugate duality in vector optimization. J. Math.Anal. Appl.,2008,343:853-865.
61S. J. Li, P. Zhao. A method of duality for a mixed vector equilibriumproblem. Optim. Lett.,2010,4:85-96.
62P. H. Sach. New generalized convexity notion for set-valued maps and appli-cation to vector optimization. J. Optim. Theory Appl.,2005,125:157-179.
63P. H. Sach. Moreau-Rockafellar Theorems for Nonconvex Set-Valued Maps.J. Optim. Theory Appl.,2007,133:213-227.
64P. Wolfe. A duality theorem for nonlinear programming. Quart. Appl.Math.,1961,19:239-244.
65B. Mond, T. Weir. Generalized concavity and duality. In: Generalizedconcavity in Optimization and Economics, S. Schaible and W.T. Ziemba,(Eds.), Academic Press, New York,1981,263-279.
66B. Mond, R. R. Egudo. On strict converse duality in nonlinear programming.J. Infor. optim. Sci.,1985,6:113-116.
67B. Mond, T. Weir, B. D. Craven. On duality in generalized minimax pro-grams. Asia-Pacifc J. Oper. Res.,1990,7:55-62.
68Z. A. Liang, H. X. Huang, P. M. Pardalos. Optimality conditions and dualityfor a class of nonlinear fractional programming problems. J. Optim. TheoryAppl.,2001,110:611-619.
69M. A. Hanson, R. Pini, C. Singh. Multiobjective programming under gen-eralized type-I invexity. J. Math. Anal. Appl.,2001,261:562-577.
70T. Antczak, Multiobjective programming under d-invexity. Eur. J. Oper.Res.,2002,137:28-36.
71Y. L. Ye. d-invexity and optimality conditions. J. Math. Anal. Appl.,1991,153:242-249.
72J. C. Liu, C. S. Wu. On minimax fractional optimality conditions withinvexity. J. Math. Anal. Appl.,1998,219:21-35.
73J. C. Liu, C. S. Wu. On minimax fractional optimality conditions with(F, ρ)-convexity. J. Math. Anal. Appl.,1998,219:36-51.
74S. K. Mishra, S. Y. Wang, K. K. Lai. Nondiferentiable multiobjective pro-gramming under generalized d-univexity. Eur. J. Oper. Res.,2005,160:218–226.
75C. Nahak, S. Nanda. Multiobjective duality with ρ (η, θ) invexity. J.Appl. Math. Stoch. Anal.,2005,2:175-180.
76R. P. Pant, J. S. Rautela. α-invexity and duality in mathematical program-ming. J. Math. Anal. Appr. Theory,2006,1(2):104-114.
77S. K. Mishra, R. P. Pant, J. S. Rautela. Generalized α-invexity and non-diferentiable minimax fractional programming. J. Comput. Appl. Math.,2007,206:122-135.
78S. K. Mishra, R. P. Pant, J. S. Rautela. Generalized α-univexity and dual-ity for nondiferentiable minimax fractional programming. Nonlinear Anal-Theor.,2009,70:144-158.
79O. L. Mangasarian. Second and higher-order duality in nonlinear program-ming. J. Math. Anal. Appl.,1975,51:607-620.
80B. Mond. Second order duality for nonlinear programs. Opsearch.,1974,11:90-99.
81C. R. Bector, B. K. Bector.(Generalized)-bonvex functions and secondorder duality for a nonlinear programming problem. Congr. Number.,1986,52:37-52.
82C. R. Bector, S. Chandra. Second order symmetric and self-dual programs.Opsearch.,1986,23:89-95.
83C. R. Bector, B. K. Bector. On various duality theorems for second orderduality in nonlinear programming. Cahiers Center Etudes Rech. Opre.,1986,28:283-792.
84C. R. Bector, S. Chandra. First and second order duality for a class ofnondiferentiable fractional programming problem. J. Infor. Optim. Sci.,1986,7:335-348.
85C. R. Bector, S. Chandra. Generalized-convexity and higher order dualityfor fractional programming. Opsearch.,1987,24:143-154.
86C. R. Bector, S. Chandra, I. Husain. Second order duality in continuoustime nonlinear programming. Congr. Number.,1987,56:121-134.
87C. R. Bector, S. Chandra, I. Husain. Second order duality for a minimaxprogramming problem. Opsearch.,1991,28:249-263.
88I. Ahmad, Z. Husain. Second order (F, α, ρ, d)-convexity and duality inmultiobjective programming. Inform. Sci.,2006,176:3094–3103.
89H.C. Lai, J.C. Liu, K. Tanaka. Necessary and sufcient conditions for min-imax fractional programming. J. Math. Anal. Appl.,1999,230:311–328.
90I. Ahmad, Z. Husain. Duality in nondiferentiable minimax fractional pro-gramming with generalized convexity. Appl. Math. Comput.,2006,176:545–551.
91Z. Husain, A. Jayswal, I. Ahmad. Second order duality for nondiferentiableminimax programming problems with generalized convexity. J. Glob. Op-tim.,2009,44:593-608.
92I. Ahmad, Z. Husain, S. AI-Homidan. Second-order duality in nondiferen-tiable fractional programming. Nonlinear Anal-Real,2011,12:1103-1110.
93A. N. Tykhonov. On the stability of the functional optimization problem.USSR Comput. Math. Math. Phys.,1966,6:28-33.
94E. S. Levitin, B. T. Polyak. Convergence of minimizing sequences in condi-tional extremum problems. Soviet Mathematics Doklady,1966,7:764-767.
95X. X. Huang, X. Q. Yang. Generalized Levitin-Polyak well-posedness inconstrained optimization. SIAM J. Optim.,2006,17(1):243-258.
96X. X. Huang, X. Q. Yang, D. L. Zhu. Levitin-Polyak well-posedness ofvariational inequality problems with functional constraints. J. Glob. Optim.,2009,44:159-174.
97B. Jiang, J. Zhang, X. X. Huang. Levitin-Polyak well-posedness of gener-alized quasivariational inequalities with functional constraints. NonlinearAnal-Theor,2009,70:1492-1503.
98T. Zolezzi. Well-posedness criteria in optimization with application to thecalculus of variations. Nonlinear Anal.,1995,25:437-453.
99T. Zolezzi. Extended well-posedness of optimization problems. J. Optim.Theory Appl.,1996,91(1):257-266.
100B. Lemaire, C. Ould Ahmed Salem, J.P. Revalski. Well-posedness by per-turbations of variational problems. J, Optim. Theory Appl.,2002,115:345-368.
101Y. P. Fang, N. J. Huang, J. C. Yao. Well-posedness of mixed variationalinequalities, inclusion problems and fxed point problems. J. Glob. Optim.,2008,41:117-133.
102D. Dentcheva, S. Helbig. On variational principles, level sets, well-posednessand ε-solutions in vector optimization. J. Optim. Theory Appl.,1996,89:329-349.
103X. X. Huang. Extended well-posedness properties of vector optimizationproblems. J. Optim. Theory Appl.,2000,106:165-182.
104X. X. Huang. Pointwise well-posedness of perturbed vector optimizationproblems in a vector-valued variational principle. J. Optim. Theory Appl.,2001,108:671-686.
105X. X. Huang, X. Q. Yang. Levitin-Polyak well-posedness of constrainedvector optimization problems. J. Glob. Optim.,2007,37:287-304.
106X. X. Huang, X. Q. Yang. Further study on the Levitin-Polyak well-posedness of constrained convex vector optimization problems. NonlinearAnal.,2012,75:1341-1347.
107G. P. Crespi, A. Guerraggio, M. Rocca. Well posedness in vector optimiza-tion problems and vector variational inequalities. J. Optim. Theory Appl.,2007,132:213-226.
108G. P. Crespi, M. Papalia, M. Rocca. Extended well-posedness of quasiconvexvector optimization problems. J. Optim. Theory Appl.,2009,141:285-297.
109M. Bianchi, G. Kassay, R. Pini. Well-posedness for vector equilibrium prob-lems. Math. Methods Oper. Res.,2009,70:171-182.
110J. W. Peng, S. Y. Wu. The generalized Tykhonov well-posedness for systemof vector quasi-equilibrium problems. Optim. Lett.,2010,4:501-512.
111X. J. Long, N. J. Huang. Metric characterizations of α-well-posedness forsymmetric quasi-equilibrium problems. J. Glob. Optim.,2009,45:459-471.
112S.J. Li, M. H. Li. Levitin-Polyak well-posedness of vector equilibrium prob-lems. Math. Methods Oper. Res.,2009,69:125-140.
113S. J. Li, W. Y. Zhang. Hadamard well-posed vector optimization problems.J. Glob. Optim.,2010,46:383-393.
114C. R. Bector, Programming Problems with Convex Fractional Functions,Indian Institute of Technology,1967,383-391.
115B.D. Craven, B. Mond. The dual of a fractional linear program. J. Math.Anal. Appl.,1973,42:507-512.
116S. Schaible. Fractional programming: applications and algorithms. Eur. J.Oper. Res.,1981,7:111-120.
117S. Chandra, V. Kumar. Duality in fractional minimax programming. J.Austral. Math. Soc. Ser. A,1995,58:376-386.
118X. M. Yang, S. H. Hou. On minimax fractional optimality and duality withgeneralized convexity, J. Glob. Optim.,2005,31(2):235-252.
119W. Dinkelbach. On nonlinear programming. Management Science,1967,13(7):526-543.
120S. S. Kutateladze. Changes of variables in the Young transformation. Sov.Math. Dokl.,1977,18(2):545-548.
121C. J. Goh, X. Q. Yang. Duality in optimization and variational inequalities.Taylor and Francis, London,2002.
122R. I. Bo t, S. M. Grad, G. Wanka. Duality in vector optimization. Springer-Verlag Berlin Heidelberg,2009.
123J. F. Bard. Some properties of the bilevel programming problem. J. Optim.Theory Appl.,1991,68:371-378.
124S. Dempe. A necessary and sufcient optimality condition for bilevel pro-gramming problem. Optimization,1992,25:341-354.
125S. Dempe. First order necessary optimality conditions for general bilevelprogramming problems. J. Optim. Theory Appl.,1997,95:735-739.
126J. J. Ye. Nondiferentiable multiplier rules for optimization and bilevel op-timization problems. SIAM J. Optim.,2004,15:252-274.
127J. J. Ye. Necessary optimality conditions for multiobjective bilevel programs.Math. Oper. Res.,2011,36(1):165-184.
128W. D. Rong, Y. N. Wu. ε-weak minimal solutions of vector optimizationproblems with set-valued maps. J. Optim. Theory Appl.,2000,106:569-579.
129D. Z. Du, P. M. Pardalos. Minimax and applications. Kluwer AcademicPublishers, The Netherlands,1995.
130P. Pandian, G. Natarajan. Second order (b; F)-convexity in multiobjectivenonlinear programming. Int. J. Math. Anal.,2010,4(7):303-314.
131J. Zhang, B. Mond. Second order duality for multiobjective nonlinear pro-gramming involving generalized convexity. In: Proceedings of the Optimiza-tion Miniconference III, B. M. Glover, B. D. Craven and D. Ralph (Eds.),University of Ballarat,1997,79-95.
132S. Chandra, V. Kumar. Duality in fractional minimax programming. J.Austral. Math. Soc. Ser. A,1995,58:376-386.
133M. B. Lignola, J. Morgan. Well-posedness for optimization problems withconstraints defned by variational inequalities having a unique solution. J.Global Optim.,2000,16(1):57-67.
134Y. P. Fang, R. Hu. Parametric well-posedness for variational inequalitiesdefned by bifunctions. Comput. Math. Appl.,2007,53:1306-1316.
135M. B. Lignola. J. Morgan, Semicontinuity and episemicontinuity: equiva-lence and applications. Boll. Unione Mat. Ital.1994,8B:1-6.
136J. Banas, K. Goebel. Measures of noncompactness in Banach spaces. MarcelDekker, New York and Basel,1980.
137J. P. Aubin. Applied nonlinear analysis. New York: Wiley,1984.