Abstract
The purpose of this paper is to estimate the approximate solutions for variational inequalities. In terms of estimate functions, we establish some estimates of the sizes of the approximate solutions from outside and inside respectively. By considering the behaviors of estimate functions, we give some characterizations of the well-posedness for variational inequalities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Auslender A (1976) Optimization, Méthodes Numériques. Masson, Paris
Del Prete I, Lignola MB, Morgan J (2003) New concepts of well-posedness for optimization problems with variational inequality constraints. JIPAM. J Inequal Pure Appl Math 4(1): Article 5
Dontchev AL, Zolezzi T (1993) Well-posed optimization problems, Lecture Notes in Math. 1543 Springer, Berlin Heidelberg New York
Huang XX (2001) Extended and strongly extended well-posedness of set-valued optimization problems. Math Methods Oper Res 53:101–116
Kinderlehrer D, Stampacchia G (1980) An introduction to variational inequalities and their applications. Academic, New York
Lignola MB, Morgan J (2000) Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution. J Global Optim 16(1):57–67
Lignola MB, Morgan J (2002) Approximating solutions and α-well-posedness for variational inequalities and Nash equilibria. In: Decision and control in management science. Kluwer, Dordrecht pp 367–378
Lignola MB (2006) Well-posedness and L-well-posedness for quasivariational inequalities. J Optim Theory Appl 128(1):119–138
Lucchetti R, Patrone F (1981) A characterization of Tyhonov well-posedness for minimum problems, with applications to variational inequalities. Numer Funct Anal Optim 3(4):461–476
Miglierina E, Molho E (2003) Well-posedness and convexity in vector optimization. Math Methods Oper Res 58:375–385
Tykhonov AN (1966) On the stability of the functional optimization problem. USSR J Comput Math Math Phys 6:631–634
Yang XM, Yang XQ, Teo KL (2004) Some remarks on the Minty vector variational inequality. J Optim Theory Appl 121(1):193–201
Zolezzi T (1995) Well-posedness criteria in optimization with application to the calculus of variations. Nonlinear Anal TMA 25:437–453
Zolezzi T (1996) Extended well-posedness of optimization problems. J Optim Theory Appl 91:257–266
Zolezzi T (2001) Well-posedness and optimization under perturbations. Optimization with data perturbations. Ann Oper Res 101:351–361
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by the Basic and Applied Research Projection of Sichuan Province (05JY029-009-1) and the National Natural Science Foundation of China (10671135).
Rights and permissions
About this article
Cite this article
Fang, YP., Hu, R. Estimates of approximate solutions and well-posedness for variational inequalities. Math Meth Oper Res 65, 281–291 (2007). https://doi.org/10.1007/s00186-006-0122-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00186-006-0122-0