Abstract
We first present a proper condition under which the image of a set-valued mapping becomes a singleton and then obtain several generic uniqueness theorems which can be applied to study the uniqueness of the solutions for nonlinear problems. As applications, we prove that, in the sense of Baire category, most optimization problems (respectively, saddle point problems and variational inequality problems) have unique solution.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aliprantis C.D., Border K.C.: Infinite Dimensional Analysis. Springer, Berlin (1999)
Beer G.: On a generic optimization theorem of Petar Kenderov. Nonlinear Anal. TMA 12, 627–655 (1988)
Bianchi M., Schaible S.: Equilibrium problems under generalized convexity and generalized monotonicity. J. Glob. Optim. 30, 121–134 (2004)
Blum E., Oettli W.: From optimization and variational inequalities to equilibrium problems. Math. Stud. 63, 123–145 (1994)
Christensen J.P.R.: Theorems of Namioka and Johnsonn type for upper semi-continuous and compact valued set-valued mappings. Proc. Am. Math. Soc. 86, 649–655 (1982)
Fort M.K.: Points of continuity of semi-continuous functions. Publ. Math. Debrecen 2, 100–102 (1951)
Kenderov P.S.: Most of the optimization problems have unique solution. In: Brosowski, B., Deutsch, F. (eds), Proceedings, Oberwolfach on Parametric Optimization. Birkhäuser International Series of Numerical Mathematics, vol. 72, pp. 203–216. Birkhäuser, Basel (1984)
Kenderov P.S., Ribarska N.K.: Most of the two person zero-sum games have unique solution, Workshop/Mini-Conference on Functional Analysis and Optimization, Canberra, pp. 73–82 (1988)
Tan K.K., Yu J., Yuan X.Z.: The stability of Ky Fan’s points. Proc. Am. Math. Soc. 123, 1511–1519 (1995)
Tan K.K., Yu J., Yuan X.Z.: The uniqueness of saddle points. Bull. Pol. Acad. Sci. Math. 43, 119–129 (1995)
Yu J.: Essential weak effficient solution in multiobjective optimization problems. J. Math. Anal. Appl. 166, 230–235 (1992)
Yu J., Peng D.T., Xiang S.W.: Generic uniqueness of equilibrium points. Nonlinear Anal. TMA 74, 6326–6332 (2011)
Zaslavski A.J.: Generic existence of a saddle point. Commun. Appl. Anal. 8, 143–151 (2004)
Zaslavski A.J.: Optimization on Metric and Normed Spaces. Springer, New York (2010)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Peng, D., Yu, J. & Xiu, N. Generic uniqueness theorems with some applications. J Glob Optim 56, 713–725 (2013). https://doi.org/10.1007/s10898-012-9903-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10898-012-9903-6