Forward–backward-based descent methods for composite variational inequalities

Abstract

We consider the monotone composite variational inequality (CVI) where the underlying mapping is formed as the sum of two monotone mappings. We combine the forward–backward and descent direction ideas together, and thus present the unified algorithmic framework of forward–backward-based descent methods for solving the CVI. A new iterate of such a method is generated by a prediction–correction fashion, where the predictor is yielded by the forward–backward method and then the predictor is corrected by a descent step. We derive some implementable forward–backward-based descent algorithms for some concrete cases of the CVI, and verify their numerical efficiency via preliminary numerical experiments.

Keywords:

• variational inequality
• composite mapping
• forward–backward method
• contraction
• descent direction

Acknowledgements

We are grateful to Prof. Y. X. Yuan and three anonymous referees for their valuable comments which have helped us improve the presentation of this paper substantially. We particularly thank Prof. I. V. Konnov for sending us some papers including 16. This work has been presented on ‘The 8th International Conference on Optimization: Techniques and Applications’ held at Shanghai, China, December 2010. B.H. was supported by the NSFC Grant 10971095 and the NSF of Jiangsu Province Grant BK2008255. X.Y. was supported by the Hong Kong General Research Fund: HKBU 203311.

Notes

We thank Dr Junfeng Yang for making us aware of the continuation technique.