Gap functions and existence of solutions to set-valued vector variational inequalities. (English) Zbl 1027.49003

Summary: The variational inequality problem with set-valued 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 set-valued 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 semi-infinite programming problem. We investigate also the existence of a solution for the generalized VVI with a set-valued mapping by virtue of the existence of a solution of the VVI with a single-valued function and a continuous selection theorem.


49J40 Variational inequalities
90C34 Semi-infinite programming
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[1] GIANNESSI F., Theorems of the Alternative, Quadratic Programs, and Complementarity Problems, Variational Inequality and Complementarity Problems, Edited by R. W. Cottle, F. Giannessi, and J. L. Lions, Wiley, New York, NY, pp. 151–186, 1980.
[2] CHEN, G. Y., and YANG, X. Q., The Vector Complementary Problem and Its Equivalence with the Weak Minimal Element in Ordered Spaces, Journal of Mathematical Analysis and Applications, Vol. 153, pp. 136–158, 1990. · Zbl 0712.90083
[3] KONNOV, I. V., and YAO, J. C., On the Generalized Vector Variational Inequality Problem, Journal of Mathematical Analysis and Applications, Vol. 206, pp. 42–58, 1997. · Zbl 0878.49006
[4] GOH, C. J., and YANG, X. Q., Vector Equilibrium Problems and Vector Optimization, European Journal of Operational Research, Vol. 116, pp. 615–628, 1999. · Zbl 1009.90093
[5] BROWDER, F. E., and HESS, P., Nonlinear Mappings of Monotone Type in Banach Spaces, Journal of Functional Analysis. Vol. 11, pp. 251–294, 1972. · Zbl 0249.47044
[6] KRAVVARITTIS, D., Nonlinear Equations and Inequalities in Banach Spaces, Journal of Mathematical Analysis and Applications, Vol. 67, pp. 205–214, 1979. · Zbl 0404.47033
[7] ANSARI, Q. H., and YAO, J. C., Nondifferentiable and Nonconuex Optimization Problems, Journal of Optimization Theory and Applications, Vol. 106, pp. 475–488, 2000. · Zbl 0970.90092
[8] CHEN, G. Y., GOH, C. J., and YANG, X. Q., On a Gap Function for Vector Variational Inequalities, Vector Variational Inequalities and Vector Equilibria, Edited by F. Giannessi, Kluwer Academic Publishers, Dordrecht, Holland, pp. 55–72, 2000. · Zbl 0997.49006
[9] AUBIN, J. P., and EKELAND, I., Applied Nonlinear Analysis, John Wiley and Sons, New York, NY, 1984. · Zbl 0641.47066
[10] 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.
[11] REEMTSEN, R., and RUCKMANN, J. J., Editors, Semi-Infinite Programming, Kluwer Academic Publishers, Dordrecht, Holland, 1998.
[12] DING, X. P., KIM, W. K., and TAN, K. K., A Selection Theorem and Its Applications, Bulletin of the Australian Mathematical Society, Vol. 46, pp. 205–212, 1992. · Zbl 0762.47030
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