Abstract
Generalized convex functions preserve many valuable properties of mathematical programming problems with convex functions. Generalized monotone maps allow for an extension of existence results for variational inequality problems with monotone maps. Both models are special realizations of an abstract equilibrium problem with numerous applications, especially in equilibrium analysis (e.g., Blum and Oettli, 1994). We survey existence results for equilibrium problems obtained under generalized convexity and generalized monotonicity. We consider both the scalar and the vector case. Finally existence results for a system of vector equilibrium problems under generalized convexity are surveyed which have applications to a system of vector variational inequality problems. Throughout the survey we demonstrate that the results can be obtained without the rigid assumptions of convexity and monotonicity.
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References
Allevi, E., Gnudi, A. and Konnov, I.V. (2001), Generalized vector variational inequalities over product sets, Nonlinear Analysis 47, 573-582.
Ansari, Q.H. and Yao, J.-C. (1999), A fixed point theorem and its applications to the system of variational inequalities, Bullettin of the Australian Mathematical Society 59, 433-442.
Q.H., Schaible, S. and Yao, J.-C. (2000), System of vector equilibrium problems and its applications, Journal of Optimization Theory and Applications 107, 547-557.
Ansari, Q.H., Schaible, S. and Yao, J.-C. (2002), The system of generalized vector equilibrium problems with applications, Journal of Global Optimization, 22, 3-16.
Avriel, M., Diewert, W.E., Schaible, S. and Zang, I. (1988), Generalized Concavity, Plenum Publishing Corporation, New York.
Bianchi, M. (1993), Pseudo P-monotone operators and variational inequalities, report No.6, Istituto di Econometria e Matematica per le Decisioni Economiche, Università Cattolica del Sacro Cuore, Milan, Italy.
Bianchi, M. and Schaible, S. (1996), Generalized monotone bifunctions and equilibrium problems, Journal of Optimization Theory and Applications 90, 31-43.
Bianchi, M., Hadjisavvas, N. and Schaible, S. (1997), Vector equilibrium problems with generalized monotone bifunctions, Journal of Optimization Theory and Applications 92, 527-542.
Blum, E. and Oettli, W. (1994) From optimization and variational inequalities to equilibrium problems, The Mathematics Student 63, 123-145.
Brezis, H., Niremberg, L. and Stampacchia, G. (1972), A remark on Fan ‘s minimax principle, Bollettino della Unione Matematica Italiana 6, 293-300.
Cambini, A., Castagnoli, E., Martein, L., Mazzoleni, P. and Schaible, S. (eds.) (1990), Generalized Convexity and Fractional Programming with Economic Applications, Volume 345 of Lecture Notes in Economics and Mathematical Systems. Springer, Berlin, Heidelberg, New York.
Cohen, G. and Chaplais, F. (1988), Nested monotonicity for variational inequalities over a product of spaces and convergence of iterative algorithms, Journal of Optimization Theory and Applications 59, 360-390.
Crouzeix, J-P., Martinez-Legaz, J.-E. and Volle, M. (eds.) (1998) Generalized Convexity, Generalized Monotonicity. Kluwer Academic Publishers, Dordrecht, Boston, London.
Deguire, P., Tan, K.K. and Yuan, G.X.-Z. (1999), The study of maximal elements, fixed points for L s -majorized mappings and their applications to minimax and variational inequalities in the product topological spaces, Nonlinear Analysis: Theory, Methods and Applications 37, 933-951.
Fan, K. (1961), A generalization of Tychonoff ‘s fixed point theorem, Mathematische Annalen 142, 305-310.
Hadjisavvas, N. and Schaible, S. (1993), On strong pseudomonotonicity and (semi) strict quasimonotonicity, Journal of Optimization Theory and Applications 79, 139-155.
Hadjisavvas, N. and Schaible, S. (1996), Quasimonotone variational inequalities in Banach spaces, Journal of Optimization Theory and Applications 90, 95-111.
Hadjisavvas, N. and Schaible, S. (1998), From scalar to vector equilibrium problems in the quasimonotone case, Journal of Optimization Theory and Applications 96, 297-309.
Hadjisavvas, N. and Schaible, S. (2001), Generalized monotone multi-valued maps, In: Floudas, C.A. and Pardalos, P.M. (eds), Encyclopedia of Optimization,Volume II (E-Integer), pp. 224-229.
Hadjisavvas, N., Matinez-Legaz, J.-E. and Penot, J.-P. (eds.) (2001), Generalized Convexity and Generalized Monotonicity. Volume 502 of Lecture Notes in Economics and Mathematical Systems. Springer, Berlin, Heidelberg, New York.
Hassouni, A. (1983). Sous-differentiels des fonctions quasiconvexes, These de troisieme cycle, Universite Paul Sabatier, Toulouse.
Jeyakumar, V., Oettli, W. and Natividad, M. (1993), A solvability theorem for a class of quasiconvex mappings with applications to optimization, Journal of Mathematical Analysis and Applications 179, 537-546.
Karamardian, S. (1976), Complementarity over cones with monotone and pseudomonotone maps, Journal of Optimization Theory and Applications 18, 445-454.
Karamardian, S. and Schaible, S. (1990), Seven kinds of monotone maps, Journal of Optimization Theory and Applications 66, 37-46.
Komlosi, S., Rapcsak, T. and Schaible, S. (eds.) (1994), Generalized Convexity, Volume 405 of Lecture Notes in Economics and Mathematical Systems. Springer, Berlin, Heidelberg, New York.
Konnov, I.V. (2001a), Combined Relaxation Methods for Variational Inequalities, Springer, Berlin, Heidelberg, New York.
Konnov, I.V. (2001b), Relatively monotone variational inequalities over product sets, Operations Research Letters 28, 21-26.
Luc, D.T. (1989), Theory of vector optimization, Volume 319 of Lecture Notes in Economics and Mathematical Systems. Springer, Berlin, Heidelberg, New York.
Oettli, W. (1997), A remark on vector-valued equilibria and generalized monotonicity, Acta Matematica Vietnamica 22, 213-221.
Pang, J.S. (1985), Asymmetric variational inequality problems over product sets, applications and iterative methods, Mathematical Programming 31, 206-219.
Schaible, S. and Ziemba, W.T. (eds.) (1981), Generalized Concavity in Optimization and Economics. Academic Press, New York.
Singh, C. and Dass, B.K. (eds) (1989), Continuous-Time, Fractional and Multiobjective Programming. Analytic Publishing Company, New Delhi.
Yao, J.C. (1994), Variational inequalities with generalized monotone operators, Mathematics of Operations Research, 19, 391-403.
Zukhovitskii, S.I., Polyak, R.A. and Primak, M.E. (1969), Two methods of search for equilibrium points of n-person concave games, Soviet Mathematics Doklady 10, 279-282.
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Bianchi, M., Schaible, S. Equilibrium Problems under Generalized Convexity and Generalized Monotonicity. J Glob Optim 30, 121–134 (2004). https://doi.org/10.1007/s10898-004-8269-9
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DOI: https://doi.org/10.1007/s10898-004-8269-9