Abstract
In this paper a new concept of well-posedness for variational inequalities and Nash equilibria, termed α-well-posedness, is presented. We give conditions under which a variational inequality is a-well-posed and we derive a result for Nash equilibria.
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References
Auslender, A., Optimisation: Méthodes Numériques, Masson, Paris, 1976.
Baiocchi, C. and Capelo, A., Variational and quasi-variational inequalities, applications to free boundary, John Wiley and Sons, New-York 1984.
Basar, T. and Olsder, G.J., Dynamic non cooperative games, Academic Press, New York, second edition, 1995.
Breton, M., Alj, A. and Haurie, A., Sequential Stackelberg equilibria in two-person games, Journal of Optimization Theory and Application, 59, pp. 71–97, 1988.
Cavazzuti, E. and Morgan, J., Well-posed saddle-point problems, Optimization theory and algorithms, Edited by J.B. Hiriart -urruty, W. Oettli, and J. Stoer, Lecture Notes in Pure and Applied Mathematics, M. Dekker, New York, Vol. 86, pp. 61–76, 1983.
Dontchev, A.L. and Zolezzi, T., Well-posed optimization problems, Lecture Notes in Mathematics, 1543, Springer-Verlag, Berlin, 1993.
Fukushima, M., Equivalent differentiable optimization problem and descent method for symmetric variational inequalities, Mathematical Programming, 53, pp. 99–110, 1992.
Fukushima, M., Merit functions for variational inequality and complementarity problems, in G. Di Pillo and F. Giannessi, eds. Nonlinear Optimization and Applications, Plenum Press, New York, pp. 155–170, 1996.
Fukushima, M., and Pang, J.-S., Minimizing and stationary sequences of merit functions for complementarity problems and variational inequalitites, Technical report, 1999.
Leitmann, G., On generalized Stackelberg strategies, Journal of Optimization theory and Application, 26, pp. 637–643, 1978.
Lignola, M.B. and Morgan, J., Approximate solutions to variational inequalities and application, Le Matematiche, XLIX, pp. 281–293, 1994.
Lignola, M.B. and Morgan, J., Well-posedness for optimization problems with constraints defined by variational inequalities having a unique solution, Journal of Global Optimization, 16, pp. 57–67, 1999.
Lucchetti, R. and Patrone, F., A characterization of Tychonov well-posedness for minimum problems, with applications to variational inequalities, Numerical Functional Analysis and Optimization, 3, pp. 461–476, 1981.
Luo, Z.Q., Pang, J.S. and Ralph, D., Mathematical Programs with Equilibrium Constraints, Cambridge University press, 1996.
Mallozzi, L. and Morgan, J. Mixed strategies for hierarchical zero-sum games, to appear on Annals of Dynamic Games.
Marcotte, P. and Zhu, D.L., Exact and inexact penalty methods for generalized bilevel programming problems, Mathematical Programming, 75, pp. 19–76, 1996.
Margiocco, M., Patrone, F. and Pusillo, L., A new approach to Tikhonov well-posedness for Nash equilibria, Optimization, 40, pp. 385–400, 1997.
Morgan, J. and Raucci, R., New convergence results for Nash equilibria, Journal of Convex Analysis, 6, n. 2, pp. 377–385, 1999.
Outrata, J.V., On optimization problems with variational inequality constraint, Siam Journal on Optimization, 4, pp. 334–357, 1994.
Outrata, J.V., Kocvara, M. and Zowe, J., Nonsmooth Approach to Optimization Problems with Equilibrium Constraints, Nonconvex Optimization and its Applications, Kluwer Academic publishers, 1998.
Revalski, J., Variational Inequalities with unique solution, in Mathematics and Education in Mathematics, Proceedings of the 14th Spring Conference of the Union of Bulgarian Mathematicians, Sofia, 1985.
Stackeiberg, H. von, Marktform und Gleichgewicht, Julius Springer, Vienna, 1934.
Ye, J.J., Zhu, D.L. and Zhu, Q., Generalized bilevel programming problems, DMS-646-IR, Department of Mathematics and Statistics, University of Victoria, 1993.
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Lignola, M.B., Morgan, J. (2002). Approximate Solutions and α-Well-Posedness for Variational Inequalities and Nash Equilibria. In: Zaccour, G. (eds) Decision & Control in Management Science. Advances in Computational Management Science, vol 4. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-3561-1_20
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DOI: https://doi.org/10.1007/978-1-4757-3561-1_20
Publisher Name: Springer, Boston, MA
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