Abstract
In this paper, we consider variational inequalities with pseudomonotone maps which depend on a parameter and study the behavior of their solutions. The main result gives sufficient conditions for the stability of the initial variational inequality problem under small perturbations of the parameter. As an application, we obtain a stability result for a class of parametric optimization problems.
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References
Hartman, G. J., and Stampacchia, G., On Some Nonlinear Elliptic Differential Functional Equations, Acta Mathematica, Vol. 115, pp. 271–310, 1966.
Kinderlehrer, D., and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, NY, 1980.
Zeidler, E., Nonlinear Functional Analysis and Its Applications, Vols. 2A and 2B, Springer Verlag, Berlin, Germany, 1990.
Karamardian, S., Complementarity Problems over Cones with Monotone and Pseudomonotone Maps, Journal of Optimization Theory and Applications, Vol. 18, pp. 445–454, 1976.
Cottle, R. W., and Yao, J. C., Pseudomonotone Complementarity Problems in Hilbert Space, Journal of Optimization Theory and Applications, Vol. 75, pp. 281–295, 1992.
Yao, J. C., Multivalued Variational Inequalities with K-Pseudomonotone Operators, Journal of Optimization Theory and Applications, Vol. 83, pp. 391–403, 1994.
Ding, X. P., and Tarafdar, E., Monotone Generalized Variational Inequalities and Generalized Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 88, pp. 107–122, 1996.
Schaible, S., and Yao, J. C., On the Equivalence of Nonlinear Complementarity Problems and Least-Element Problems, Mathematical Programming, Vol. 70, pp. 191–200, 1995.
Hadjisavvas, N., and Schaible, S., Quasimonotone Variational Inequalities in Banach Spaces, Journal of Optimization Theory and Applications, Vol. 90, 1996.
Karamardian, S., and Schaible, S., Seven Kinds of Monotone Maps, Journal of Optimization Theory and Applications, Vol. 66, pp. 37–46, 1990.
Avriel, M., Dievert, W. E., Schaible, S., and Zang, I., Generalized Concavity, Plenum Publishing Corporation, New York, NY, 1988.
Dafermos, S., Sensitivity Analysis in Variational Inequalities, Mathematics of Operations Research, Vol. 13, pp. 421–434, 1988.
Kassay, G., and KolumbÁn, I., Implicit Function Theorems for Monotone Mappings, Preprint 7, Babeş-Bolyai University Cluj, Romania, pp. 1–27, 1988.
Kassay, G., and KolumbÁn, I., Implicit Functions and Variational Inequalities for Monotone Mappings, Preprint 7, Babeş-Bolyai University Cluj, Romania, pp. 79–92, 1989.
Alt, W., and KolumbÁn, I., An Implicit-Function Theorem for a Class of Monotone Generalized Equations, Kybernetika, Vol. 29, pp. 210–221, 1993.
Martos, B., Nonlinear Programming: Theory and Methods, Akadémiai Kiadó, Budapest, Hungary, 1975.
Karamardian, S., Schaible, S., and Crouzeix, J. P., Characterizations of Generalized Monotone Maps, Journal of Optimization Theory and Applications, Vol. 76, pp. 399–413, 1993.
Aubin, J. P., and Frankowska, H., Set-Valued Analysis, Birkhaüser, Boston, Massachusetts, 1990.
Robinson, S. M., Regularity and Stability for Convex Multivalued Functions, Mathematics of Operations Research, Vol. 1, pp. 130–143, 1976.
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Kassay, G., Kolumban, J. Multivalued Parametric Variational Inequalities with α-Pseudomonotone Maps. Journal of Optimization Theory and Applications 107, 35–50 (2000). https://doi.org/10.1023/A:1004600631797
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DOI: https://doi.org/10.1023/A:1004600631797