Abstract
Necessary and sufficient conditions for a point to be a weak saddle point of a vector valued function (i.e. to be a solution of the vector saddle point problem) are given. Also, an existence result for a vector saddle point to have a solution is given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ansari Q.H., Yao J.C.: An existence result for the generalized vector equilibrium problem. Appl. Math. Lett. 12, 53–56 (1999)
Chen G.Y.: Existence of solutions for a vector variational inequality: An extension of the Hartman-Stampacchia theorem. J. Optim. Theory Appl. 74(3), 445–456 (1992)
Chinchuluun, A., Pardalos, P., Migdalas, A., Pitsoulis, L. (eds): Pareto otimality, game theory and equilibria. Springer, Berlin (2008)
Duca D.I., Lupşa L.: Bi-(φ,ψ) convex sets. Math. Pannonica 14(2), 193–203 (2003)
Duca D.I., Lupşa L.: Bi-(φ,ψ) convex–concave functions. Buletinul Stiinţific al Universită ţii “Politehnica” din Timişoara 48(62), 67–72 (2003)
Duca D.I., Lupşa L.: On the E-epigraph of an E-convex function. J. Optim. Theory Appl. 129(2), 341–348 (2006)
Duca D.I., Duca E.: Parametrized saddle points. Acta Math. Vietnam. 35(3), 411–426 (2010)
Ehrgott M., Wiecek M.M.: Saddle points and Pareto points in multiple objective programming. J. Glob. Optim. 32(1), 11–33 (2005)
Fan K.: A generalization of Tychonoff’s fixed point theorem. Math. Ann. 142, 305–310 (1961)
Ferro F.: A minimax theorem for vector-valued functions. J. Optim. Theory Appl. 60(1), 19–31 (1989)
Kazmi K.R., Khan S.: Existence of solutions for a vector saddle point problem. Bull. Austral. Math. Soc. 61, 201–206 (2000)
Kimura, K., Tanaka, T.: Existence theorems of saddle points for vector valued functions. In: Takahashi, W., and Tanaka, T. (eds.) Proceedings of the International Conference on Nonlinear Analysis and Convex Analysis. Yokohama Publishers, pp. 169–178 (2003)
Lin L.-J.: Existence theorems of simultaneous equilibrium problems and generalizad vector quasi-saddle points. J. Glob. Optim. 32(4), 613–632 (2005)
Nieuwenhuis J.W.: Some minimax theorems in vector-valued functions. J. Optim. Theory Appl. 40(3), 463–475 (1983)
Pardalos P.M., Rassias T.M., Khan A.A.: Nonlinear analysis and variational problems. Springer, Berlin (2010)
Noor M.A., Rassias T.M.: On nonconvex equilibrium problems. J. Math. Anal. Appl. 312, 289–299 (2005)
Tanaka T.: Generalized quasiconvexities, cone saddle points, and minimax theorem for vector-valued functions. J. Optim. Theory Appl. 81(2), 355–377 (1994)
Tanaka T.: Generalized semicontinuity and existence theorems for cone saddle points. Appl. Math. Optim. 36, 313–322 (1997)
Yang X.-M.: On E-convex sets, E-convex functions, and E-convex programming. J. Optim. Theory Appl. 109(3), 699–704 (2001)
Youness E.A.: E-convex sets, E-convex functions, and E-convex programming. J. Optim. Theory Appl. 102(2), 439–450 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Duca, D.I., Lupsa, L. Saddle points for vector valued functions: existence, necessary and sufficient theorems. J Glob Optim 53, 431–440 (2012). https://doi.org/10.1007/s10898-011-9721-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-011-9721-2
Keywords
- Convex function
- Vector-valued function
- Weak saddle point
- Vector saddle point problem
- Vector variational inequlitie