A hybrid splitting method for variational inequality problems with separable structure

Abstract

Alternating direction method (ADM), which decomposes a large-scale original variational inequality (VI) problem into a series of smaller scale subproblems, is very attractive for solving a class of VI problems with a separable structure. This type of method can greatly improve the efficiency, but cannot avoid solving VI subproblems. In this paper, we propose a hybrid splitting method with variable parameters for separable VI problems. Specifically, the proposed method solves only one strongly monotone VI subproblem and a well-posed system of nonlinear equations in each iteration. The global convergence of the new method is established under some standard assumptions as those in classical ADMs. Finally, some preliminary numerical results show that the proposed method performs favourably in practice.

Keywords:

• splitting method
• alternating direction method
• nonlinear equations
• variational inequality problems
• separable structure

AMS Subject Classifications:

• 65K05
• 90C30

Acknowledgements

The authors gratefully acknowledge the anonymous referees for their useful suggestions and comments to improve this paper. This work was supported by the National Natural Science Foundation of China Nos 10871098, 11071122, 11071123, 11171159; the Specialized Research Fund of Doctoral Programme of Higher Education of China No. 20103207110002 and the Natural Science Fund of Jiangsu Province No. BK2009397.