Outer approximation methods for solving variational inequalities in Hilbert space

Abstract

In this paper, we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set C. We assume that the set C can be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method, the main idea of which is to project at each step onto a particular half-space constructed using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. In the present paper, we establish strong convergence in Hilbert space. We emphasize that to the best of our knowledge, Fukushima’s method has so far been considered only in the Euclidean setting with different conditions on F. We provide several examples for the case where C is the common fixed point set of a finite number of cutters with numerical illustrations of our theoretical results.

Keywords:

• Common fixed point
• iterative method
• quasi-nonexpansive operator
• subgradient projection
• variational inequality

Acknowledgements

We are grateful to the anonymous referee for his/her comments and remarks.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported in part by the Israel Science Foundation [grant number 389/12]; the Fund for the Promotion of Research at the Technion and by the Technion General Research Fund. The third author was financially supported by the Polish National Science Centre within the framework of the Etiuda funding scheme under agreement number [DEC-2013/08/T/ST1/00177].