Abstract
In this paper, the existence of the solution for strong vector equilibrium problems is studied by using the separation theorem for convex sets. The arc-wise connectedness and the closedness of the strong solution set for vector equilibrium problems are discussed; and a necessary and sufficient condition for the strong solution is obtained.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Blum E., Oettli W. (1994). From optimization and variational inequalities to equilibrium problems. Math. Student 63:123–145
Ansari, Q.H., Oettli, W., Schläger, D.: A generalization of vectorial equilibria. In: Operations Research and Its Applications, Second International Symposium, ISORA’96 Guilin, P.R. China, December 11–14, 1996 Proceedings, World Publishing Corporation, 181–185 (1996)
Giannessi, F., Mastroeni, G., Pellegrini, L.: On the theory of vector optimization and variational inequalities. Image space analysis and separation. In: Giannessi (ed.) Vector Variational Inequalities and Vector Equilibria, Mathematical Theories, pp. 153–215. Kluwer Academic Publishers (2000)
Gong X.H. (2001). Efficiency and Henig efficiency for vector equilibrium problems. J. Optim. Theor. Appl 108:139–154
Gong, X.H., Fu, W.T., Liu, W.: Super efficiency for a vector equilibrium in locally convex topological vector spaces. In: Giannessi (ed.) Vector Variational Inequalities and Vector Equilibria, Mathematical Theories, pp. 233–252. Kluwer Academic Publishers (2000)
Martellotti A., Salvadori A. (1990). Inequality systems and minimax results without linear structure. J. Math. Anal. Appl. 148:79–86
Fu J.Y. (2000). Generalized vector quasi-equilibrium problem. Math. Methods Operation Res. 52:57–64
Chen G.Y., Yang X.Q. (1990). Vector complimentarity problem and its equivalence with weak minimal element in ordered spaces. J. Math. Anal. Appl. 153:136–158
Chen G.Y. (1992). Existence of solution for a vector variational inequality: an extension of the Hartman-Stampacchia theorem. J. Optim. Theor. Appl. 74:445–456
Yang X.Q. (1993). Vector varitional inequality and its duality. Nonlinear Anal. Theor. Methods Appl. 21:869–877
Siddiqi A.H., Ansari Q.H., Khaliq A. (1995). On vector variational inequalities. J. Optim. Theor. Appl. 84:171–180
Chen G.Y., Li S.J. (1996). Existence of solution for a generalized vector quasivariational Inequality. J. Optim. Theor. Appl. 90:321–334
Yu S.J., Yao J.C. (1996). On vector variational inequalities. J. Optim. Theor. Appl. 89:749–769
Bianch M., Hadjisavvas N., Schaible S. (1997). Vector equilibrium problems with generalized monotone bifunctions. J. Optim. Theor. Appl. 92:527–542
Konnov I.V., Yao J.C. (1999). Existence of solutions for generalized vector equilibrium problems. J. Math. Anal. Appl. 233: 328–335
Lee G.M., Lee B.S., Chang S.S. (1996). On vector quasivariational inequalities. J. Math. Anal. Appl. 203:626–638
Giannessi, F. (ed.): Vector Variational Inequalities and Vector Eqilibria. Mathematical Theories. Kluwer Publishers (2000)
Ansari Q.H., Schaible S., Yao J.C. (2000). System of vector equilibrium problems and its applications. J. Optim. Theor. Appl. 107:547–557
Mastroeni G., Rapcsák T. (2000). On convex generalized systems. J. Optim. Theor. Appl. 104:605–627
Bianchi M., Pini R. (2002). A result on localization of equilibria. J. Optim. Theor. Appl. 115:335–343
Fabián F.B., Frnando F.B. (2003). Vector equilibrium problems under asymptotic analysis. J. Global Optim. 26:141–166
Lin L.J., Ansari Q.H., Wu J.Y. (2003). Geometric properties and coincidence theorems with applications to generalized vector equilibrium problems. J. Optim. Theor. Appl. 117:121–137
Li J., Huang N.J., Kim J.K. (2003). On implicit vector equilibrium problems. J. Math. Anal. Appl. 283:501–512
Fu J.Y. (2003). Symmetric vector quasi-equilibrium problems. J. Math. Anal. Appl. 285:708–713
Jameson G. (1970). Ordered Linear Spaces, Lecture Notes in Math. Springer-Verlag, Berlin, 141p.
Martellotti A., Salvadori A. (1988). A minimax theorem for functions taking values in a Riesz space. J. Math. Anal. Appl. 133: 1–13
Fan, K.: Minimax Theorem. National Academy of Sciences, Washington, DC, Proceedings USA 39, 42–47 (1953)
Stoer J., Witzgall C. (1970). Convexity and Optimization in Finite Dimensions: I. Springer-Verlag, New York
Jeyakumar V. (1986). A generalization of a minimax theorem of Fan via a theorem of the alternative. J. Optim. Theor. Appl. 48:525–533
Gwinner J., Oettli W. (1994). Theorems of the alternative and duality for inf-sup problems. Math. Operations Res. 19:238–256
Luc, D.T.: Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems. Spring-Verlag (1989)
Ferro F. (1989). A minimax theorem for vector-valued functions. J. Optim. Theor. Appl. 60:19–31
Jahn J. (1986). Mathematical Vector Optimization in Partially Ordered Linear Spaces. Verlag Peter Lang, New York
Aubin J.P., Ekeland I. (1984). Applied Nonlinear Analysis. John Wiley, New York
Jahn J., Rauh R. (1997). Contingent epiderivatives and set-valued optimization. Math. Methods Operation Res. 46:193–211
Gong X.H., Dong H.B., Wang S.Y. (2003). Optimality conditions for proper efficient solutions of vector set-valued optimization. J. Math. Anal. Appl. 284:332–350
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xunhua, G. Strong Vector Equilibrium Problems. J Glob Optim 36, 339–349 (2006). https://doi.org/10.1007/s10898-006-9012-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-006-9012-5