Extragradient method for solving quasivariational inequalities

Abstract

We study methods for solving a class of the quasivariational inequalities in Hilbert space when the changeable set is described by translation of a fixed, closed and convex set. We consider one variant of the gradient-type projection method and an extragradient method. The possibilities of the choice of parameters of the gradient projection method in this case are wider than in the general case of a changeable set. The extragradient method on each iteration makes one trial step along the gradient, and the value of the gradient at the obtained point is used at the first point as the iteration direction. In the paper, we establish sufficient conditions for the convergence of the proposed methods and derive a new estimate of the rates of the convergence. The main result of this paper is contained in the convergence analysis of the extragradient method.

Keywords:

• Gradient projection method
• extragradient method
• quasivariational inequalities
• rate of convergence

Acknowledgements

We are very grateful to the two referees for their valuable comments on the paper, which have considerably improved the paper.

Notes

No potential conflict of interest was reported by the authors.