×
Ansari, Q. H.; Konnov, I. V.; Yao, J. C.

Characterizations of solutions for vector equilibrium problems. (English) Zbl 1012.90055

J. Optimization Theory Appl. 113, No. 3, 435-447 (2002).
Summary: We characterize the solutions of vector equilibrium problems as well as dual vector equilibrium problems. We establish also vector optimization problem formulations of set-valued maps for vector equilibrium problems and dual vector equilibrium problems, which include vector variational inequality problems and vector complementarity problems. The set-valued maps involved in our formulations depend on the data of the vector equilibrium problems, but not on their solution sets. We prove also that the solution sets of our vector optimization problems of set-valued maps contain or coincide with the solution sets of the vector equilibrium problems.

Cited in 50 Documents

MSC:

90C29 Multi-objective and goal programming
91A40 Other game-theoretic models
90C47 Minimax problems in mathematical programming

Keywords:

vector optimization problems; vector equilibrium problem; dual vector equilibrium problems; set-valued maps; characterization of solutions
Cite Review PDF
Full Text: DOI

References:

[1] BLUM, E., and OETTLI, W., Variational Principles for Equilibrium Problems, Parametric Optimization and Related Topics III, Edited by J. Guddad et al., Peter Lang, Frankfurt am Main, Germany, pp. 79-88, 1993. · Zbl 0839.90016
[2] BLUM, E., and OETTLI, W., From Optimization and Variational Inequalities to Equilibrium Problems, The Mathematics Student, Vol. 63, pp. 123-145, 1994. · Zbl 0888.49007
[3] 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
[4] 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
[5] AUSLENDER, A., Optimisation: Méthodes Numériques, Masson, Paris, France, 1976.
[6] 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
[7] CHEN, G. Y., GOH, C. J., and YANG, X. Q., On Gap Functions for Vector Variational Inequalities, Vector Variational Inequalities Vector Equilibria: Mathematical Theories, Edited by F. Giannessi, Kluwer Academic Publishers, Dordrecht, Netherlands, pp. 55-72, 2000. · Zbl 0997.49006
[8] 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.
[9] 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, Netherlands, pp. 1-15, 2000. · Zbl 0992.49012
[10] 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
[11] 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
[12] FU, J., Simultaneous Vector Variational Inequalities and Vector Implicit Complementarity Problems, Journal of Optimization Theory and Applications, Vol. 93, pp. 141-151, 1997. · Zbl 0901.90169 · doi:10.1023/A:1022653918733
[13] HADJISAVVAS, N., and SCHAILE, S., From Scalar to Vector Equilibrium Problems in the Quasimonotone Case, Journal of Optimization Theory and Applications, Vol. 96, pp. 297-305, 1998. · Zbl 0903.90141 · doi:10.1023/A:1022666014055
[14] KONNOV, I. V., Combined Relaxation Method for Solûing Vector Equilibrium Problems, Russian Mathematics (Izvestiya Vuzov Math), Vol. 39, pp. 51-59, 1995. · Zbl 0887.65057
[15] LEE, G. M., KIM, D. S., and LEE, B. S., On Noncooperatiûe Vector Equilibrium, Indian Journal of Pure and Applied Mathematics, Vol. 27, pp. 735-739, 1996. · Zbl 0858.90141
[16] OETTLI, W., A Remark on Vector-Valued Equilibria and Generalized Monotonicity, Acta Mathematica Vietnamica, Vol. 22, pp. 213-221, 1997. · Zbl 0914.90235
[17] OETTLI, W., and SCHLÄGER, D., Generalized Vectorial Equilibria and Generalized Monotonicity, Functional Analysis with Current Applications in Science, Technology, and Industry, 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
[18] 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
[19] 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
[20] 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
[21] 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
[22] LUC, D. T., Theory of Vector Optimization, Lecture Notes in Economics and Mathematical System, 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.