Abstract
In this paper, two concepts of well-posedness for quasivariational inequalities having a unique solution are introduced. Some equivalent characterizations of these concepts and classes of well-posed quasivariational inequalities are presented. The corresponding concepts of well-posedness in the generalized sense are also investigated for quasivariational inequalities having more than one solution
Similar content being viewed by others
References
A Bensoussan JL. Lions (1973) ArticleTitle Controle Impulsionel et Inequations Quasivariationelles d’Evolution Comptes Rendus de l’Académie des Sciences de Paris. 276 1333–1338 Occurrence Handle47 #5692
Mosco, U., Implicit Variational Problems and Quasivariational Inequalities, Summer School, Nonlinear Operators and the Calculus of Variations, Bruxelles, Belgium, 1975; Lecture Notes in Mathematics, Springer Verlag, Berlin, Germany, Vol. 543, pp. 83-156, 1976.
C. Baiocchi A. Capelo (1984) Variational and Quasivariational Inequalities: Applications to Free Boundary Problems John Wiley and Sons New York, NY
F. Facchinei J. S. Pang (2003) Finite-Dimensional Variational Inequalities and Complementarity Problems, Springer Series in Operations Research Springer Verlag Berlin, Germany
M. Koĉvara J. Outrata (1995) ArticleTitle On a Class of Quasivariational Inequalities Optimization Methods and Software. 5 275–295
M. A. Noor A. Z. Memon (2002) ArticleTitle Algorithms for General Mixed Quasivariational Inequalities Journal of Inequalities in Pure and Applied Mathematics. 3 1–9 Occurrence Handle2003e:49020
M. B. Lignola J. Morgan (2000) ArticleTitle Well-Posedness for Optimization Problems with Constraints Defined by a Variational Inequality Having a Unique Solution Journal of Global Optimization. 16 57–67 Occurrence Handle10.1023/A:1008370910807 Occurrence Handle2001d:49040
A. L. Dontchev T. Zolezzi (1993) Well-Posed Optimization Problems, Lectures Notes in Mathematics Springer Verlag Berlin, Germany
R. Lucchetti F. Patrone (1981) ArticleTitle A Characterization of Tykhonov Well-Posedness for Minimum Problems, with Applications to Variational Inequalities Numerical Functional Analysis and Optimization. 3 461–476 Occurrence Handle82m:49036
J. Revalski (1985) Variational Inequalities with Unique Solution, Mathematics and Mathematical Education (Sunny Beach, Bulgaria, 1985) Bulgarian Academy of Sciences Sofia, Bulgaria 534–541
P. T. Harker J. S. Pang (1990) ArticleTitle Finite-Dimensional Variational Inequality and Nonlinear Complementarity Problems: A Survey of Theory, Algorithms, and Applications Mathematical Programming. 48 161–220 Occurrence Handle91g:90166
I. Del Prete M. B. Lignola J. Morgan (2003) ArticleTitle New Concepts of Well-Posedness for Optimization Problems with Variational Inequality Constraints Journal of Inequalities in Pure and Applied Mathematics 4 26–43 Occurrence Handle2004c:49061
A. Auslender (1976) Optimisation: Méthodes Numériques Masson Paris, France
M. Fukushima (1992) ArticleTitle Equivalent Differentiable Optimization Problem and Descent Method for Symmetric Variational Inequalities Mathematical Programming. 53 99–110 Occurrence Handle10.1007/BF01585696 Occurrence Handle0756.90081 Occurrence Handle92k:90100
Lignola, M. B., and Morgan, J., Approximate Solutions to Variational Inequalities and Applications: Equilibrium Problems with Side Constraints, Lagrangian Theory and Duality, Acireale, Italy, 1994, Le Mathematiche (Catania), Vol. 49, pp. 281--293, 1995.
MB. Lignola J. Morgan (2002) Approximate Solutions and α-Well-Posedness for Variational Inequalities and Nash Equilibria, Decision and Control in Management Science Kluwer Academic Publishers Dordrecht, Netherlands 367–378
M. B. Lignola J. Morgan (1994) ArticleTitle Semicontinuity and Episemicontinuity: Equivalence and Applications Bollettino dell’ Unione Matematica Italiana. 8 1–6 Occurrence Handle95c:49018
K. Kuratowski (1968) Topology Academic Press New York NY
J. Morgan M. Romaniello (2003) ArticleTitle Generalized Quasivariational Inequalities and Duality Journal of Inequalities in Pure and Applied Mathematics. 4 32–38 Occurrence Handle2004e:49010
Author information
Authors and Affiliations
Additional information
Communicated by J. P. Crouzeix
The author is grateful to an anonymous referee for valuable comments.
Rights and permissions
About this article
Cite this article
Lignola, M.B. Well-Posedness and L-Well-Posedness for Quasivariational Inequalities. J Optim Theory Appl 128, 119–138 (2006). https://doi.org/10.1007/s10957-005-7561-2
Issue Date:
DOI: https://doi.org/10.1007/s10957-005-7561-2