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A projection-proximal point algorithm for solving generalized variational inequalities. (English) Zbl 1242.90267
The study of variational inequality problems provides a nice challenge in Mathematics. This paper deals with the folllowing generalized variational inequality problem:
Let \(X\) be a nonempty, closed and convex subset of a Hilbert space \(H\) and \(T:\rightrightarrows H\) be a set-valued mapping. We consider the following generalized variational inequality: find \(x^*\in X\) and \(w^*\in T(x^*)\) such that \[ \langle w^*,y- x^*\rangle\leq 0,\quad\forall y\in X. \] In order to solve it, the authors consider a projection-proximal point method, and investigate a general iterative algorithm, which consists of an inexact proximal point step followed by a suitable orthogonal projection onto a hyperplane. When \(T\) is pseudo-monotone (in the sense of Karamardian) with weakly upper semicontinuity and weakly compact and convex values, the convergence of such an algorithm is proved. In addition, the convergence rate of the iterative sequence under suitable conditions is also analyzed.

90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
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