Gap function for set-valued vector variational-like inequalities. (English) Zbl 1146.58009

Mishra, S. K.; Wang, S. Y.; Lai, K. K.

J. Optim. Theory Appl. 138, No. 1, 77-84 (2008).

Summary: 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.

MSC:

58E35	Variational inequalities (global problems) in infinite-dimensional spaces
90C22	Semidefinite programming
49J40	Variational inequalities
49J53	Set-valued and variational analysis

Keywords:

vector variational-like inequalities; set-valued mappings; gap functions; existence of a solution; semidefinite programming

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