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Convergence Analysis of the Generalized Splitting Methods for a Class of Nonconvex Optimization Problems

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Abstract

In this paper, we propose generalized splitting methods for solving a class of nonconvex optimization problems. The new methods are extended from the classic Douglas–Rachford and Peaceman–Rachford splitting methods. The range of the new step-sizes even can be enlarged two times for some special cases. The new methods can also be used to solve convex optimization problems. In particular, for convex problems, we propose more relax conditions on step-sizes and other parameters and prove the global convergence and iteration complexity without any additional assumptions. Under the strong convexity assumption on the objective function, the linear convergence rate can be derived easily.

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Acknowledgements

The authors are grateful to thank two anonymous referees for their helpful suggestions on improving the quality of the original manuscript. This work was supported by National Natural Science Foundation of China Grants 11771078 and 71661147004, Natural Science Foundation of Jiangsu Province Grant BK20181258 and Project 333 of Jiangsu Province Grant BRA2018351.

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Authors and Affiliations

  1. School of Management and Engineering, Nanjing University, Nanjing, 210093, China

    Min Li

  2. School of Economics and Management, Southeast University, Nanjing, 210096, China

    Zhongming Wu

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  1. Min Li
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  2. Zhongming Wu
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Correspondence to Min Li.

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Communicated by Alexander S. Strekalovsky.

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Li, M., Wu, Z. Convergence Analysis of the Generalized Splitting Methods for a Class of Nonconvex Optimization Problems. J Optim Theory Appl 183, 535–565 (2019). https://doi.org/10.1007/s10957-019-01564-1

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  • DOI: https://doi.org/10.1007/s10957-019-01564-1

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