Abstract
Recently, the alternating direction method of multipliers has attracted great attention. For a class of variational inequalities (VIs), this method is efficient, when the subproblems can be solved exactly. However, the subproblems could be too difficult or impossible to be solved exactly in many practical applications. In this paper, we propose an inexact method for structured VIs based on the projection and contraction method. Instead of solving the subproblems exactly, we use the simple projection to get a predictor and correct it to approximate the subproblems’ real solutions. The convergence of the proposed method is proved under mild assumptions and its efficiency is also verified by some numerical experiments.
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Acknowledgments
The authors are very grateful to the editor and the anonymous referees for their valuable suggestions and constructive comments, which have considerably improved the presentation of the paper. Qingzhi Yang’s work is supported by the National Natural Science Foundation of China (Grant No. 11271206), Doctoral fund of Chinese Ministry of Education (Grant No. 20120031110024).
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Chen, Z., Wan, L. & Yang, Q. An Inexact Alternating Direction Method for Structured Variational Inequalities. J Optim Theory Appl 163, 439–459 (2014). https://doi.org/10.1007/s10957-014-0522-x
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DOI: https://doi.org/10.1007/s10957-014-0522-x