A partial inexact alternating direction method for structured variational inequalities

Abstract

In this article, a new method is proposed for solving a class of structured variational inequalities (SVIs). The proposed method is referred to as the partial inexact proximal alternating direction (piPAD) method. In the method, two subproblems are solved independently. One is handled by an inexact proximal point method and the other is solved directly. This feature is the major difference between the proposed method and some existing alternating direction-like methods. The convergence of the piPAD method is proved. Two examples of the modern convex optimization problem arising from engineering and information sciences, which can be reformulated into the encountered SVIs, are presented to demonstrate the applicability of the piPAD method. Also, some preliminary numerical results are reported to validate the feasibility and efficiency of the piPAD method.

Keywords:

• structured variational inequalities
• separable convex optimization
• alternating direction method
• inexact proximal point algorithm

Acknowledgements

The authors would like to thank the anonymous referees as well as the associate editor for their constructive comments, which has improved the quality of our manuscript.

This work was supported partially by the National Science Foundation of China (61170308), the Natural Science Foundation of FuJian Province (2011J01008), and the FuJian Province Education Department (JA11033).