Abstract
In this paper, we consider some well-known equilibrium problems and their duals in a topological Hausdorff vector space X for a bifunction F defined on K x K,where K is a convex subset of X. Some necessary conditions are investigated, proving different results depending on the behaviour of F on the diagonal set. The concept of proper quasimonotonicity for bifunctions is defined, and the relationship with generalized monotonicity is investigated. The main result proves that the condition of proper quasimonotonicity is sharp in order to solve the dual equilibrium problem on every convex set.
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References
Bianchi, M. and Schaible, S. (1996), Generalized Monotone Bifunctions and Equilibrium Problems, J. Optim. Theory Appl. 90: 31–43.
Blum, E. and Oettli, W. (1994), From Optimization and Variational Inequalities to Equilibrium Problems, The Mathematics Student 63: 123–145.
Chadli, O., Chbani, Z. and Rihai, H. (2000) Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities, J. Optim. Theory Appl. 105: 299–323.
Daniilidis, A. and Hadjisavvas, N. (2000), On Generalized Cyclically Monotone Operators and Proper Quasimonotonicity, Optimization 47: 123–135.
Dugundji, J. and Granas, A. (1978), KKM Maps and Variational Inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5(4): 679–682.
Fan, K. (1961), A Generalization of Tychonoff's Fixed point Theorem, Math. Ann. 142: 305–310
Giannessi, F. (1997), On Minty Variational Principle, in: F. Giannessi, S. Komlosi and T. Rapsack (eds), New Trend in Mathematical Programming, Kluwer Academic Publishers, Dordrecht, The Netherlands.
Harker, P.T. and Pang, J.S. (1990), Finite Dimensional Variational Inequalities and Nonlinear Complementarity Problems: a Survey of Theory. Algorithms and Applications, Math. Programming 48: 161–220.
John, R. (1999), A Note on Minty Variational Inequalities and Generalized Monotonicity (preprint).
Komlosi, S. (1999), On the Stampacchia and Minty Variational Inequalities, in: G. Giorgi and F. Rossi (eds), Generalized Convexity and Optimization for Economic and Financial Decisions, Pitagora Editrice.
Komlosi, S. (1995), Generalized Monotonicity and Generalized Convexity, J. Optim. Theory Appl. 84: 361–376.
Komlosi, S. (1994), Generalized Monotonicity in Non-Smooth Analysis, in: S. Komlosi, T. Rapcsak and S. Schaible (eds), Generalized Convexity, Springer-Verlag.
Martos, B. (1975), Nonlinear Programming Theory and Methods, North Holland, Amsterdam.
Mosco, U. (1976), Implicit Variational Problems and Quasivariational Inequalities, Lectures Notes in Math. 543: 83–156 Springer-Verlag.
Rockafellar, R.T. (1972), Convex Analysis, Princeton University Press, Princeton, New Jersey.
Tan, K.K. and Yuan, Z.Z. (1993), A Minimax Inequality with Applications to Existence of Equilibrium Points, Bull. Austral. Math. Soc. 47: 483–503.
Zhou, J.X. and Chen, G. (1998), Diagonal Convexity Conditions for Problems in Convex Analysis and Quasi-Variational Inequalities, J. Math. Anal. Appl. 132: 213–225.
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Bianchi, M., Pini, R. A Note on Equilibrium Problems with Properly Quasimonotone Bifunctions. Journal of Global Optimization 20, 67–76 (2001). https://doi.org/10.1023/A:1011234525151
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DOI: https://doi.org/10.1023/A:1011234525151