×

Existence of a solution and variational principles for vector equilibrium problems. (English) Zbl 0988.49004

Summary: In this paper, we prove an existence result for a solution to the vector equilibrium problems. Then, we establish variational principles, that is, vector optimization formulations of set-valued maps for vector equilibrium problems. A perturbation function is involved in our variational principles. We prove also that the solution sets of our vector optimization problems of set-valued maps contain or coincide with the solution sets of vector equilibrium problems.

MSC:

49J40 Variational inequalities
49J53 Set-valued and variational analysis
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Kinderlehrer, D., and Stampacchia, G., An Introduction to Variational Inequalities and Their Applications, Academic Press, New York, NY, 1980. · Zbl 0457.35001
[2] Giannessi, F., Theorems of the Alternative, Quadratic Programs, and Complementarity Problems, Variational Inequalities and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, John Wiley and Sons, New York, NY, pp. 151–186, 1980.
[3] Giannessi, F., Editor, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Kluwer Academic Publishers, Dordrecht, Holland, 2000. · Zbl 0952.00009
[4] Chen, G. Y., Goh, C. J., and Yang, X. Q., On Gap Functions for Vector Variational Inequalities, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Edited by F. Giannessi, Kluwer Academic Publishers, Dordrecht, Holland, pp. 55–72, 2000. · Zbl 0997.49006
[5] Ansari, Q. H., Wong, N. C., and Yao, J. C., The Existence of Nonlinear Inequalities, Applied Mathematical Letters, Vol. 12, pp. 89–92, 1999. · Zbl 0940.49010 · doi:10.1016/S0893-9659(99)00062-2
[6] Antipin, A. S., On Convergence of Proximal Methods to Fixed Points of Extremal Mappings and Estimates of Their Rate of Convergence, Computational Mathematics and Mathematical Physics, Vol. 35, pp. 539–551, 1995. · Zbl 0852.65046
[7] Aubin, J. P., L’Analyse Non Linéaire et Ses Motivations Économiques, Masson, Paris, France, 1984.
[8] Bianchi, M., and Schaible, S., Generalized Monotone Bifunctions and Equilibrium Problems, Journal of Optimization Theory and Applications, Vol. 90, pp. 31–43, 1996. · Zbl 0903.49006 · doi:10.1007/BF02192244
[9] Blum, E., and Oettli, W., From Optimization and Variational Inequalities to Equilibrium Problems, Mathematics Student, Vol. 63, pp. 123–145, 1994. · Zbl 0888.49007
[10] BŔezis, H., Nirenberg, L., and Stampacchia, G., A Remark on Ky Fan’s Minimax Principle, Bolletino della Unione Matematica Italiana, Vol. 6, pp. 293–300, 1972. · Zbl 0264.49013
[11] Chadli, O., Chbani, Z., and Riahi, H., Recession Methods for Equilibrium Problems and Applications to Variational and Hemivariational Inequalities, Discrete and Continuous Dynamical Systems, Vol. 5, pp. 185–195, 1999. · Zbl 0949.49008
[12] Chadli, O., Chbani, Z., and Riahi, H., Equilibrium Problems and Noncoercive Variational Inequalities, Optimization, Vol. 49, pp. 1–12, 1999. · Zbl 1022.49013
[13] Chadli, O., Chbani, Z., and Riahi, H., Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities, Journal of Optimization Theory and Applications, Vol. 105, pp. 299–323, 2000. · Zbl 0966.91049 · doi:10.1023/A:1004657817758
[14] Hadjisavvas, N., and Schaible, S., From Scalar to Vector Equilibrium Problems in the Quasimonotone Case, Journal of Optimization Theory and Applications, Vol. 96, pp. 297–309, 1998. · Zbl 0903.90141 · doi:10.1023/A:1022666014055
[15] Hadjisavvas, N., and Schaible, S., Quasimonotonicity and Pseudomonotonicity in Variational Inequalities and Equilibrium Problems, Generalized Convexity, Generalized Monotonicity: Recent Results, Edited by J. P. Crouzeix, J. E. Martinez-Legaz, and M. Volle, Kluwer Academic Publishers, Dordrecht, Holland, pp. 257–275, 1998. · Zbl 0946.49005
[16] Husain, T., and Tarafdar, E., Simultaneous Variational Inequalities, Minimization Problems, and Related Results, Mathematica Japonica, Vol. 39, pp. 221– 231, 1994. · Zbl 0802.47059
[17] Konnov, I. V., A Generalized Approach to Finding a Stationary Point and the Solution of Related Problems, Computational Mathematics and Mathematical Physics, Vol. 36, pp. 585–593, 1996. · Zbl 1161.90491
[18] Tarafdar, E., and Yuan, G. X. Z., Generalized Variational Inequalities and Their Applications, Nonlinear Analysis: Theory, Methods, and Applications, Vol. 30, pp. 4171–4181, 1997. · Zbl 0912.49004 · doi:10.1016/S0362-546X(96)00142-3
[19] Yuan, G. X. Z., KKM Theory and Applications in Nonlinear Analysis, Marcel Dekker, New York, NY, 1999. · Zbl 0936.47034
[20] Ansari, Q. H., Vector Equilibrium Problems and Vector Variational Inequalities, Vector Variational Inequalities and Vector Equilibria: Mathematical Theories, Edited by F. Giannessi, Kluwer Academic Publishers, Dordrecht, Holland, pp. 1–15, 2000. · Zbl 0992.49012
[21] Ansari, Q. H., Oettli, W., and SchlÄger, D., A Generalization of Vectorial Equilibria, Mathematical Methods of Operations Research, Vol. 46, pp. 147–152, 1997. · Zbl 0889.90155 · doi:10.1007/BF01217687
[22] Bianchi, M., Hadjisavvas, N., and Schaible, S., Vector Equilibrium Problems with Generalized Monotone Bifunctions, Journal of Optimization Theory and Applications, Vol. 92, pp. 527–542, 1997. · Zbl 0878.49007 · doi:10.1023/A:1022603406244
[23] Konnov, I. V., Combined Relaxation Method for Solving Vector Equilibrium Problems, Russian Mathematics, Vol. 39, pp. 51–59, 1995. · Zbl 0887.65057
[24] Lee, G. M., Kim, D. S., and Lee, B. S., On Noncooperative Vector Equilibrium, Indian Journal of Pure and Applied Mathematics, Vol. 27, pp. 735–739, 1996. · Zbl 0858.90141
[25] Oettli, W., A Remark on Vector-Valued Equilibria and Generalized Monotonicity, Acta Mathematica Vietnamica, Vol. 22, pp. 213–221, 1997. · Zbl 0914.90235
[26] Oettli, W., and SchlÄger, D., Generalized Vectorial Equilibria and Generalized Monotonicity, Functional Analysis with Current Applications in Science, Technology, and Industries, Edited by M. Brokate and A. H. Siddiqi, Pitman Research Notes in Mathematics, Longman, Essex, England, Vol. 377, pp. 145–154, 1998. · Zbl 0904.90150
[27] Tan, N. X., and Tinh, P. N., On the Existence of Equilibrium Points of Vector Functions, Numerical Functional Analysis and Optimization, Vol. 19, pp. 141– 156, 1998. · Zbl 0896.90161
[28] Auchmuty, G., Variational Principles for Variational Inequalities, Numerical Functional Analysis and Optimization, Vol. 10, pp. 863–874, 1989. · Zbl 0678.49010 · doi:10.1080/01630568908816335
[29] Auslender, A., Optimisation: Méthodes Numériques, Masson, Paris, France, 1976.
[30] Hearn, D. W., The Gap Function of a Convex Program, Operations Research Letters, Vol. 1, pp. 67–71, 1982. · Zbl 0486.90070 · doi:10.1016/0167-6377(82)90049-9
[31] Blum, E., and Oettli, W., Variational Principles for Equilibrium Problems, Parametric Optimization and Related Topics III, Edited by J. Guddat et al., Peter Lang, Frankfurt am Main, Germany, pp. 79–88, 1993. · Zbl 0839.90016
[32] Ansari, Q. H., Konnov, I. V., and Yao, J. C., Characterizations of Solutions for Vector Equilibrium Problems, Journal of Optimization Theory and Applications (to appear). · Zbl 1012.90055
[33] Jeykumar, V., Oettli, W., and Natividad, M., A Solvability Theorem for a Class of Quasiconvex Mappings with Applications to Optimization, Journal of Mathematical Analysis and Applications, Vol. 179, pp. 537–546, 1993. · Zbl 0791.46002 · doi:10.1006/jmaa.1993.1368
[34] Tanaka, T., Generalized Quasiconvexities, Cone Saddle Points, and Minimax Theorem for Vector-Valued Functions, Journal of Optimization Theory and Applications, Vol. 81, pp. 355–377, 1994. · Zbl 0826.90102 · doi:10.1007/BF02191669
[35] Fan, K., A Generalization of Tichonoff’s Fixed-Point Theorem, Mathematische Annalen, Vol. 142, pp. 305–310, 1961. · Zbl 0093.36701 · doi:10.1007/BF01353421
[36] Chowdhury, M. S. R., and Tan, K. K., Generalized Variational Inequalities for Quasimonotone Operators and Applications, Bulletin of the Polish Academy of Sciences, Mathematics, Vol. 45, pp. 25–54, 1997. · Zbl 0885.47018
[37] Corley, H. W., Existence and Lagrange Duality for Maximization of Set-Valued Functions, Journal of Optimization Theory and Applications, Vol. 54, pp. 489–501, 1987. · Zbl 0595.90085 · doi:10.1007/BF00940198
[38] Corley, H. W., Optimality Conditions for Maximization of Set-Valued Functions, Journal of Optimization Theory and Applications, Vol. 58, pp. 1–10, 1988. · Zbl 0956.90509 · doi:10.1007/BF00939767
[39] Li, Z. F., and Chen, G. Y., Lagrangian Multipliers, Saddle Points, and Duality in Vector Optimization of Set-Valued Maps, Journal of Mathematical Analysis and Applications, Vol. 215, pp. 297–316, 1997. · Zbl 0893.90150 · doi:10.1006/jmaa.1997.5568
[40] Lin, L. J., Optimization of Set-Valued Functions, Journal of Mathematical Analysis and Applications, Vol. 186, pp. 30–51, 1994. · Zbl 0987.49011 · doi:10.1006/jmaa.1994.1284
[41] Luc, D. T., Theory of Vector Optimization, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, New York, NY, Vol. 319, 1989.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.