Gap Function for Set-Valued Vector Variational-Like Inequalities
- 142 Downloads
Variational-like inequalities with set-valued mappings are very useful in economics and nonsmooth optimization problems. In this paper, we study the existence of solutions and the formulation of solution methods for vector variational-like inequalities (VVLI) with set-valued mappings. We introduce gap functions and establish necessary and sufficient conditions for the existence of a solution of the VVLI. We investigate the existence of a solution for the generalized VVLI with a set-valued mapping by exploiting the existence of a solution of the VVLI with a single-valued function and a continuous selection theorem.
KeywordsVector variational-like inequalities Set-valued mappings Gap functions Existence of a solution Semidefinite programming
- 6.Chen, G.-Y., Cheng, G.M.: Vector variational inequalities and vector optimization. In: Lecture Notes in Economics and Mathematical Systems, vol. 285, pp. 408–416. Springer, Berlin (1987) Google Scholar
- 10.Chen, G.-Y., Goh, C.J., Yang, X.Q.: On a gap function for vector variational inequalities. In: Giannessi, F. (ed.) Vector Variational Inequalities and Vector Equilibria, pp. 55–72. Kluwer Academic, Dordrecht (2000) Google Scholar
- 12.Ding, X.P., Tarafdar, E.: Generalized vector variational-like inequalities with C x-η-pseudo-monotone set-valued mapping. In: Giannessi, F. (ed.) Vector variational inequalities and vector equilibria, pp. 125–140. Kluwer Academic, Dordrecht (2000) Google Scholar