- Research Article
- Open Access
Kuhn-Tucker Optimality Conditions for Vector Equilibrium Problems
© Z.-F.Wei and X.-H. Gong. 2010
- Received: 20 January 2010
- Accepted: 17 April 2010
- Published: 23 May 2010
By using the concept of Fréchet differentiability of mapping, we present the Kuhn-Tucker optimality conditions for weakly efficient solution, Henig efficient solution, superefficient solution, and globally efficient solution to the vector equilibrium problems with constraints.
- Convex Cone
- Efficient Solution
- Vector Optimization Problem
- Vector Variational Inequality
- Vector Equilibrium Problem
Recently, some authors have studied the optimality conditions for vector variational inequalities. Giannessi et al.  turned the vector variational inequalities with constraints into vector variational inequalities without constraints and gave sufficient conditions for the efficient solutions and the weakly efficient solutions to the vector variational inequalities in . By using the concept of subdifferential of the function, Morgan and Romaniello  gave the scalarization and Kuhn-Tucker-like conditions for the weak vector generalized quasivariational inequalities in Hilbert space. Yang and Zheng  gave the optimality conditions for approximate solutions of vector variational inequalities in Banach space. On the other hand, some authors have derived the optimality conditions for weakly efficient solutions to vector optimization problems (see [4–19]).
Vector variational inequality problems and vector optimization problems, as well as several other problems, are special realizations of vector equilibrium problems (see [20, 21]); therefore, it is important to give the optimality conditions for the solution to the vector equilibrium problems for in this way we can turn the vector equilibrium problem with constraints to a corresponding scalar optimization problem without constraints, and we can then determine if the solution of the scalar optimization problem is a solution of the original vector equilibrium problem. Under the assumption of convexity, Gong  investigated optimality conditions for weakly efficient solutions, Henig solutions, superefficient solutions, and globally efficient solutions to vector equilibrium problems with constraints and obtained that the weakly efficient solutions, Henig efficient solutions, globally efficient solutions, and superefficient solutions to vector equilibrium problems with constraints are equivalent to solution of corresponding scalar optimization problems without constraints, respectively. Qiu  presented the necessary and sufficient conditions for globally efficient solution under generalized cone-subconvexlikeness. Gong and Xiong  weakened the convexity assumptions in  and obtained necessary and sufficient conditions for weakly efficient solution, too.
In this paper, by using the concept of Fréchet differentiability of mapping, we study the optimality conditions for weakly efficient solutions, Henig solutions, superefficient solutions, and globally efficient solutions to the vector equilibrium problems. We give Kuhn-Tucker necessary conditions to the vector equilibrium problems without convexity conditions and Kuhn-Tucker sufficient conditions with convexity conditions.
Throughout this paper, let be a real normed space, let and be real partially ordered normed spaces, and let and be closed convex pointed cones with , where denotes the interior of the set .
Denote the closure of by and interior of by .
A nonempty convex subset of the convex cone is called a base of , if and . It is easy to see that if and only if has a base.
By the separation theorem of convex sets, we know .
Denote the closed unit ball of by . Suppose that has a base . Let . It is clear that . The will be used for the rest of the paper. For any , denote cone , then cl is a closed convex pointed cone, and , for all (see ).
Let be a nonempty open convex subset, and let , be mappings.
where is a convex cone in .
is called a weakly efficient solution to the VEPC.
Definition 2.2 (see ).
Definition 2.3 (see ).
Definition 2.4 (see ).
where is closed unit ball of .
exists, then is called the Gâteaux derivative of at in the direction . If this limit exists for each direction , the mapping is called Gâteaux differentiable at .
then is called the Fréchet derivative of at and is called Fréchet differentiable at .
By [26, Lemma .18], we can see that if is Fréchet differentiable at , then is Gâteaux differentiable at and the Fréchet derivative of at is equal to the Gâteaux derivative of at in each direction .
Lemma 2.9 (see ).
Assume that pointed convex cone has a base ; then one has the following.
(i)For any , .
(ii)For any , there exists some with .
(iii) , and when is bounded and closed, then , where is the interior of in with respect to the norm of .
In this section, we give the Kuhn-Tucker necessary conditions and Kuhn-Tucker sufficient conditions for weakly efficient solution, Henig efficient solution, globally efficient solution, and superefficient solution to the vector equilibrium problems with constraints.
By the proof of Theorem .20 in  or from the definition, we have the following lemma.
for all , ,
This completes the proof.
The condition that there exists such that is a generalized Slater constraint condition. In fact, from the proof of Theorem 3.2, we obtain that there exists such that and .
then is a weakly efficient solution to the VEPC.
This contradicts (3.30). Hence, is a weakly efficient solution to the VEPC.
From Theorems 3.2 and 3.4, and Remark 2.7, we get the following corollary.
The concept of weakly efficient solution to the vector equilibrium problem requires the condition that the ordering cone has an nonempty interior. If the ordering cone has an empty interior, we cannot discuss the property of weakly efficient solutions to the vector equilibrium problem. However, if the ordering cone has a base, we can give necessary conditions and sufficient conditions for Henig efficient solutions and globally efficient solutions to the vector equilibrium problems with constraints.
This completes the proof.
then is a Henig efficient solution to the VEPC.
This contradicts (3.45). Hence, is a Henig efficient solution to the VEPC.
If has a bounded closed base , in view of Lemma 2.9, we have ; besides, is a superefficient solution to the VEPC if and only if is a Henig efficient solution to the VEPC (see ). Hence, by Corollary 3.8, we have the following corollary.
for all , ,
This completes the proof.
then is a globally efficient solution to the VEPC.
This contradicts (3.65). Hence, is a globally efficient solution to the VEPC.
This research was partially supported by the National Natural Science Foundation of China and the Natural Science Foundation of Jiangxi Province (2008GZS0072), China.
- Giannessi F, Mastroeni G, Pellegrini L: On the theory of vector optimization and variational inequalities. Image space analysis and separation. In Vector Variational Inequalities and Vector Equilibria, Nonconvex Optimization and Its Applications. Volume 38. Edited by: Giannessi F. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:153–215. 10.1007/978-1-4613-0299-5_11View ArticleGoogle Scholar
- Morgan J, Romaniello M: Scalarization and Kuhn-Tucker-like conditions for weak vector generalized quasivariational inequalities. Journal of Optimization Theory and Applications 2006, 130(2):309–316. 10.1007/s10957-006-9104-xMathSciNetView ArticleMATHGoogle Scholar
- Yang XQ, Zheng XY: Approximate solutions and optimality conditions of vector variational inequalities in Banach spaces. Journal of Global Optimization 2008, 40(1–3):455–462.MathSciNetView ArticleMATHGoogle Scholar
- Mordukhovich BS, Treiman JS, Zhu QJ: An extended extremal principle with applications to multiobjective optimization. SIAM Journal on Optimization 2003, 14(2):359–379. 10.1137/S1052623402414701MathSciNetView ArticleMATHGoogle Scholar
- Ye JJ, Zhu QJ: Multiobjective optimization problem with variational inequality constraints. Mathematical Programming 2003, 96(1):139–160. 10.1007/s10107-002-0365-3MathSciNetView ArticleMATHGoogle Scholar
- Zheng XY, Ng KF: The Fermat rule for multifunctions on Banach spaces. Mathematical Programming 2005, 104(1):69–90. 10.1007/s10107-004-0569-9MathSciNetView ArticleMATHGoogle Scholar
- Mordukhovich BS: Variational Analysis and Generalized Differentiation. I. Basic Theory, Grundlehren der Mathematischen Wissenschaften. Volume 330. Springer, Berlin, Germany; 2006:xxii+579.View ArticleGoogle Scholar
- Mordukhovich BS: Variational Analysis and Generalized Differentiation. II. Applications, Grundlehren der Mathematischen Wissenschaften. Volume 331. Springer, Berlin, Germany; 2006:i–xxii and 1–610.View ArticleGoogle Scholar
- Dutta J, Tammer C: Lagrangian conditions for vector optimization in Banach spaces. Mathematical Methods of Operations Research 2006, 64(3):521–540. 10.1007/s00186-006-0079-zMathSciNetView ArticleMATHGoogle Scholar
- Zheng XY, Yang XM, Teo KL: Super-efficiency of vector optimization in Banach spaces. Journal of Mathematical Analysis and Applications 2007, 327(1):453–460. 10.1016/j.jmaa.2006.04.052MathSciNetView ArticleMATHGoogle Scholar
- Huang H: The Lagrange multiplier rule for super efficiency in vector optimization. Journal of Mathematical Analysis and Applications 2008, 342(1):503–513. 10.1016/j.jmaa.2007.12.027MathSciNetView ArticleMATHGoogle Scholar
- Mordukhovich BS: Methods of variational analysis in multiobjective optimization. Optimization 2009, 58(4):413–430. 10.1080/02331930701763660MathSciNetView ArticleMATHGoogle Scholar
- Corley HW: Optimality conditions for maximizations of set-valued functions. Journal of Optimization Theory and Applications 1988, 58(1):1–10. 10.1007/BF00939767MathSciNetView ArticleMATHGoogle Scholar
- Jahn J, Rauh R: Contingent epiderivatives and set-valued optimization. Mathematical Methods of Operations Research 1997, 46(2):193–211. 10.1007/BF01217690MathSciNetView ArticleMATHGoogle Scholar
- Chen GY, Jahn J: Optimality conditions for set-valued optimization problems. Mathematical Methods of Operations Research 1998, 48(2):187–200. 10.1007/s001860050021MathSciNetView ArticleMATHGoogle Scholar
- Liu W, Gong X-H: Proper efficiency for set-valued vector optimization problems and vector variational inequalities. Mathematical Methods of Operations Research 2000, 51(3):443–457. 10.1007/PL00003994MathSciNetView ArticleMATHGoogle Scholar
- Song J, Dong H-B, Gong X-H: Proper efficiency in vector set-valued optimization problem. Journal of Nanchang University 2001, 25: 122–130.Google Scholar
- Gong X-H, Dong H-B, Wang S-Y: Optimality conditions for proper efficient solutions of vector set-valued optimization. Journal of Mathematical Analysis and Applications 2003, 284(1):332–350. 10.1016/S0022-247X(03)00360-3MathSciNetView ArticleMATHGoogle Scholar
- Taa A: Set-valued derivatives of multifunctions and optimality conditions. Numerical Functional Analysis and Optimization 1998, 19(1–2):121–140. 10.1080/01630569808816819MathSciNetView ArticleMATHGoogle Scholar
- Gianness F (Ed): Vector Variational Inequalities and Vector Equilibria, Nonconvex Optimization and Its Applications. Volume 38. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:xiv+523.Google Scholar
- Huang NJ, Li J, Thompson HB: Stability for parametric implicit vector equilibrium problems. Mathematical and Computer Modelling 2006, 43(11–12):1267–1274. 10.1016/j.mcm.2005.06.010MathSciNetView ArticleMATHGoogle Scholar
- Gong X-H: Optimality conditions for vector equilibrium problems. Journal of Mathematical Analysis and Applications 2008, 342(2):1455–1466.MathSciNetView ArticleMATHGoogle Scholar
- Qiu Q: Optimality conditions of globally efficient solution for vector equilibrium problems with generalized convexity. Journal of Inequalities and Applications 2009, 2009:-13.Google Scholar
- Gong X-H, Xiong S-Q: The optimality conditions of the convex-like vector equilibrium problems. Journal of Nanchang University (Science Edition) 2009, 33: 409–414.Google Scholar
- Borwein JM, Zhuang D: Super efficiency in vector optimization. Transactions of the American Mathematical Society 1993, 338(1):105–122. 10.2307/2154446MathSciNetView ArticleMATHGoogle Scholar
- Jahn J: Mathematical Vector Optimization in Partially Ordered Linear Spaces, Methoden und Verfahren der Mathematischen Physik. Volume 31. Peter D. Lang, Frankfurt am Main, Germany; 1986:viii+310.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.