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A customized Douglas–Rachford splitting algorithm for separable convex minimization with linear constraints

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Abstract

We consider applying the Douglas–Rachford splitting method (DRSM) to the convex minimization problem with linear constraints and a separable objective function. The dual application of DRSM has been well studied in the literature, resulting in the well known alternating direction method of multipliers (ADMM). In this paper, we show that the primal application of DRSM in combination with an appropriate decomposition can yield an efficient structure-exploiting algorithm for the model under consideration, whose subproblems could be easier than those of ADMM. Both the exact and inexact versions of this customized DRSM are studied; and their numerical efficiency is demonstrated by some preliminary numerical results.

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Acknowledgments

The authors are grateful to two anonymous referees for their valuable comments on earlier versions of this paper; especially for one referee bringing our attention to the relevant references [13, 42]. This work was also partially supported by Institute of Computational and Theoretical Studies of Hong Kong Baptist University while the first author was a visiting research fellow of this institute.

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Authors and Affiliations

  1. School of Mathematical Sciences, Jiangsu Key Labratory for NSLSCS, Nanjing Normal University, Nanjing, 210023, China

    Deren Han

  2. School of Science, Hangzhou Dianzi University, Hangzhou, 310018, China

    Hongjin He

  3. Department of Civil Engineering, The Hong Kong University of Science and Technology, Hong Kong, China

    Hai Yang

  4. Department of Mathematics, Institute of Computational and Theoretical Studies, Hong Kong Baptist University, Hong Kong, China

    Xiaoming Yuan

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  1. Deren Han
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  2. Hongjin He
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  3. Hai Yang
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  4. Xiaoming Yuan
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Corresponding author

Correspondence to Xiaoming Yuan.

Additional information

D. Han was supported by the National Natural Science Foundation of China No. 11071122. H. He was supported by the Research Foundation of Hangzhou Dianzi University at Grant No. KYS075612037. X. Yuan was supported by the General Research Fund from Hong Kong Research Grants Council: 203712.

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Han, D., He, H., Yang, H. et al. A customized Douglas–Rachford splitting algorithm for separable convex minimization with linear constraints. Numer. Math. 127, 167–200 (2014). https://doi.org/10.1007/s00211-013-0580-2

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