Gap Functions and Error Bound to Set-Valued Variational Inequalities

A vector variational inequality (for short,VVI) in a finite-dimensional Euclidean space was introduced first by Giannessi [1] in 1980.In the recent years, variational inequalities have become a very popular field of research in optimization theory. From the computational point of view, one important research direction in variational inequalities is the study of gap functions. One advantage of the introduction of gap functions in variational inequality is that variational inequalities can be transformed into optimization problem. Thus, powerful optimization solution methods and algorithms can be applied to find solutions of variational inequalities. Meanwhile, the gap functions can be used to devise error bounds for variational inequalities. There have been many results regarding gap functions and error bounds for classical variational inequalities see[2-5].

 In this paper, we are interested in studying variational inequalities with set-valued maps. The solution of the variational inequalities with set-valued maps is a natural extension of the classic generalized variational inequalities studied in[6,7]. These kind of variational inequalities arise as generalization of optimality conditions for a constrained optimization problem with non-smooth objective function. See for example[8], where some equivalence of some particular set-valued VVI and a nondifferentiable and nonconvex vector optimization problem is established. Our aim in this article is to construct gap functions which can be used to devise error bounds for variational inequalities with set-valued maps.