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A Proximal Alternating Direction Method of Multipliers for DC Programming with Structured Constraints

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Abstract

In this paper, we consider a class of structured DC programming, where the objective function is the difference of two (possibly nonsmooth) convex functions, and the constraint is a linear function belonging to a nonempty closed convex set. To fully exploit the favorable structure of the problem under consideration, we propose an implementable algorithm, proximal Alternating Direction Method of Multipliers (pADMM), which employs the Fenchel-Young inequality and Moreau decomposition theorem such that the potentially explicit proximal operators of the two DC parts could be efficiently explored, thereby making all subproblems quite easy in some cases. Theoretically, we prove that the sequence generated by our algorithm pADMM converges to a critical point of the problem with the help of Kurdyka–Łojasiewicz inequality. Finally, some preliminary computational results on solving \(\ell _1-\alpha \ell _2\)-norm Dantzig selector problem and automated model selection demonstrate that our proposed method runs faster than the standard ADMM solver.

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Acknowledgements

The authors are grateful to the referees for their close reading and valuable comments, which have greatly helped us improve the quality of this paper. They also would like to thank Dr. Peng Li for kindly sharing his code of [16] with us, and Xiaofan Lu who shares code of [32].

Funding

H. He was supported in part by National Natural Science Foundation of China (NSFC) Grant #12371303, Zhejiang Provincial Natural Science Foundation of China (No. LZ24A010001), and Ningbo Natural Science Foundation (Project ID: 2023J014). L. Zhang was supported by NSFC Grant #12101342.

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  1. School of Mathematics and Statistics, Ningbo University, Ningbo, 315211, China

    Yingxin Zhou, Hongjin He & Linan Zhang

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  1. Yingxin Zhou
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Correspondence to Hongjin He.

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Zhou, Y., He, H. & Zhang, L. A Proximal Alternating Direction Method of Multipliers for DC Programming with Structured Constraints. J Sci Comput 99, 89 (2024). https://doi.org/10.1007/s10915-024-02550-0

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