Gap Functions and Existence of Solutions to SetValued Vector Variational Inequalities
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
The variational inequality problem with setvalued mappings is very useful in economics and nonsmooth optimization. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational inequalities (VVI) with setvalued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVI. It is shown that the optimization problem formulated by using gap functions can be transformed into a semiinfinite programming problem. We investigate also the existence of a solution for the generalized VVI with a setvalued mapping by virtue of the existence of a solution of the VVI with a singlevalued function and a continuous selection theorem.
 Giannessi, F. Theorems of the Alternative, Quadratic Programs, and Complementarity Problems. In: Cottle, R. W., Giannessi, F., Lions, J. L. eds. (1980) Variational Inequality and Complementarity Problems. Wiley, New York, NY, pp. 151186
 Chen, G. Y., Yang, X. Q. (1990) The Vector Complementary Problem and Its Equivalence with the Weak Minimal Element in Ordered Spaces. Journal of Mathematical Analysis and Applications 153: pp. 136158
 Konnov, I. V., Yao, J. C. (1997) On the Generalized Vector Variational Inequality Problem. Journal of Mathematical Analysis and Applications 206: pp. 4258
 Goh, C. J., Yang, X. Q. (1999) Vector Equilibrium Problems and Vector Optimization. European Journal of Operational Research 116: pp. 615628
 Browder, F. E., Hess, P. (1972) Nonlinear Mappings of Monotone Type in Banach Spaces. Journal of Functional Analysis 11: pp. 251294
 Kravvarittis, D. (1979) Nonlinear Equations and Inequalities in Banach Spaces. Journal of Mathematical Analysis and Applications 67: pp. 205214
 Ansari, Q. H., Yao, J. C. (2000) Nondifferentiable and Nonconuex Optimization Problems. Journal of Optimization Theory and Applications 106: pp. 475488
 Chen, G. Y., Goh, C. J., Yang, X. Q. On a Gap Function for Vector Variational Inequalities. In: Giannessi, F. eds. (2000) Vector Variational Inequalities and Vector Equilibria. Kluwer Academic Publishers, Dordrecht, Holland, pp. 5572
 Aubin, J. P., Ekeland, I. (1984) Applied Nonlinear Analysis. John Wiley and Sons, New York, NY
 GOH, C. J., and YANG, X. Q., On the Solution of a Vector Variational Inequality, Proceedings of the 4th International Conference on Optimization Techniques and Applications, Edited by L. Caccetta et al., pp. 1548–1164, 1998.
 Reemtsen, R., Ruckmann, J. J. E. eds. (1998) SemiInfinite Programming. Kluwer Academic Publishers, Dordrecht, Holland
 Ding, X. P., Kim, W. K., Tan, K. K. (1992) A Selection Theorem and Its Applications. Bulletin of the Australian Mathematical Society 46: pp. 205212
 Title
 Gap Functions and Existence of Solutions to SetValued Vector Variational Inequalities
 Journal

Journal of Optimization Theory and Applications
Volume 115, Issue 2 , pp 407417
 Cover Date
 20021101
 DOI
 10.1023/A:1020844423345
 Print ISSN
 00223239
 Online ISSN
 15732878
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 Vector variational inequalities
 setvalued mappings
 gap functions
 existence of a solution
 semiinfinite programming
 Industry Sectors