Abstract
In this paper, we present sufficient conditions for the existence of Henig efficient solutions, superefficient solutions and Henig globally efficient solutions of a vector equilibrium problem in topological vector spaces, using a well-known separation theorem in infinite dimensional spaces. As an application, using a scalarization technique, existence results for proper efficient solutions of generalized vector variational inequalities are given.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ansari Q.H., Konnov I.V., Yao J.C.: Existence of solutions and variational principles for vector equilibrium problems. J. Optim. Theory Appl. 110, 481–492 (2001)
Ansari Q.H., Konnov I.V., Yao J.C.: Characterizations of solutions for vector equilibrium problems. J. Optim. Theory Appl. 113, 435–447 (2002)
Ansari Q.H., Oettli W., Schläger D.: generalization of vector equilibria. Math. Methods Oper. Res. 46, 147–152 (1997)
Bianchi M., Hadjisavvas N., Schaible S.: Vector equilibrium problems with generalized monotone bifunctons. J. Optim. Theory Appl. 92, 527–542 (1997)
Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Student 63, 123–145 (1994)
Capătă, A., Kassay, G.: On vector equilibrium problems and applications. Taiwanese J. Math. (to appear)
Ceng L.C., Chen G.Y., Huang X.X., Yao J.C.: Existence theorems for generalized vector variational inequalities with pseudomonotonicity and their applications. Taiwanese. J. Math. 12, 151–172 (2008)
Chadli O., Wong N.C., Yao J.C.: Equilibrium problems with applications to eigenvalue problems. J. Optim. Theory Appl. 117, 245–266 (2003)
Chadli O., Schaible S., Yao J.C.: Regularized equilibrium problems with an application to noncoercive hemivariational inequalities. J. Optim. Theory. Appl. 121, 571–596 (2004)
Chadli O., Liu Z.H., Yao J.C.: Applications of equilibrium problems to a class of noncoercive variational inequalities. J. Optim. Theory Appl. 132, 89–110 (2007)
Chen G.Y., Cheng Q.M.: Vector variational inequality and vector optimization. In: Sawaragi, Y., Inoue, K., Nakayama, H. (eds) Towards Interactive and Intelligent Decision Support Systems, Series: Lecture Notes in Economical Mathematical Systems 285, pp. 408–416. Springer, New York (1987)
Chen G.Y., Hou S.H.: Existence of solutions for vector variational inequalities. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria, Series: Mathematical Theories, pp. 73–86. Kluwer, Dordrecht (2000)
Chinchuluun, A., Pardalos, P.M., Migdalas, A., Pitsoulis, L. (eds): Pareto Optimality, Game Theory and Equilibria Series: Optimization and its Applications. Vol. 17. Springer, New York (2008)
Fan K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142, 305–310 (1961)
Giannessi F.: Theorems of alternative, quadratic programs and complementary problems. In: Cottle, R.W., Giannessi, F., Lions, J.L. (eds) Variational Inequality and Complementary Problems, pp. 151–186. Wiley, New York (1980)
Göpfert A., Riahi H., Tammer C., Zălinescu C.: Variational Methods in Partially Ordered Spaces. Series: CMS Books in Mathematics 17. Springer, New York (2003)
Gong X.H., Fu W.T., Liu W.: Super efficiency for a vector equilibrium problem in locally convex topological vector spaces. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria, Series: Mathematical Theories., pp. 233–252. Kluwer, Dordrecht (2000)
Gong X.H.: Efficiency and Henig efficiency for vector equilibrium problems. J. Optim. Theory Appl. 108, 139–154 (2001)
Gong X.H.: Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior. J. Math. Anal. Appl. 307, 12–31 (2005)
Gong X.H.: Strong vector equilibrium problems. J. Global Optim. 36, 339–349 (2006)
Gong X.H.: Connectedness of the solution sets and scalarization for vector equilibrium problems. J. Optim. Theory Appl. 133, 151–161 (2007)
Gong X.H.: Optimality conditions for vector equilibrium problems. J. Math. Anal. Appl. 342, 1455–1466 (2008)
Gong X.H., Yao J.C.: Connectedness of the set of efficient solutions for generalized systems. J. Optim. Theory Appl. 138, 189–196 (2008)
Gong X.H., Yao J.C.: Lower semicontinuity of the set of efficient solutions for generalized systems. J. Optim. Theory Appl. 138, 197–205 (2008)
Kassay G., Kolumbán J.: On a generalized sup-inf problem. J. Optim. Theory Appl. 91, 651–670 (1996)
Luc D.T.: Theory of Vector Optimization Series: Lecture Notes in Economics and Mathematical Systems. Springer, Berlin (1989)
Megginson R.E.: An Introduction to Banach Space Theory Series: Graduate Texts in Mathematics 183. Springer, Berlin (1998)
Rudin, W.: Functional analysis. In: International Series in Pure and Applied Mathematics. Tata McGraw-Hill Publishing Company LTD, New-Delhi (1973)
Tan N.X., Tinh P.N.: On the existence of equilibrium points of vector functions. Numer. Funct. Anal. Optim. 19, 141–156 (1998)
Tanaka T.: Generalized semicontinuity and existence theorems for cone saddle points. Appl. Math. Optim. 36, 313–322 (1997)
Yu S.J., Yao J.C.: On vector variational inequalities. J. Optim. Theory Appl. 89, 749–769 (1996)
Zeng L.C., Wu S.Y, Yao J.C.: Generalized KKM theorem with applications to generalized minimax inequalities and generalized equilibrium problems. Taiwanese J. Math. 10, 1497–1514 (2006)
Zheng X.Y.: The domination property for efficiency in locally convex spaces. J. Math. Anal. Appl. 213, 455–467 (1997)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Capătă, A. Existence results for proper efficient solutions of vector equilibrium problems and applications. J Glob Optim 51, 657–675 (2011). https://doi.org/10.1007/s10898-011-9649-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-011-9649-6