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Solving saddle point problems: a landscape of primal-dual algorithm with larger stepsizes

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Abstract

We consider a class of saddle point problems frequently arising in the areas of image processing and machine learning. In this paper, we propose a simple primal-dual algorithm, which embeds a general proximal term induced with a positive definite matrix into one subproblem. It is remarkable that our algorithm enjoys larger stepsizes than many existing state-of-the-art primal-dual-like algorithms due to our relaxed convergence-guaranteeing condition. Moreover, our algorithm includes the well-known primal-dual hybrid gradient method as its special case, while it is also of possible benefit to deriving partially linearized primal-dual algorithms. Finally, we show that our algorithm is able to deal with multi-block separable saddle point problems. In particular, an application to a multi-block separable minimization problem with linear constraints yields a parallel algorithm. Some computational results sufficiently support the promising improvement brought by our relaxed requirement.

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Acknowledgements

The authors are grateful to two anonymous referees for their valuable comments on this paper; especially for one referee bringing our attention to the relevant reference [23]. This research was partially supported by the National Natural Science Foundation of China (Nos. 11301123, 11771113, 12201309) and the Startup Foundation for Introducing Talent of NUIST (No. 2022r027).

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Authors and Affiliations

  1. Department of Information and Computing Science, School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Fan Jiang

  2. School of Mathematical Sciences, Xiamen University, Xiamen, 360015, China

    Zhiyuan Zhang

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

    Hongjin He

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  1. Fan Jiang
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  2. Zhiyuan Zhang
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Correspondence to Hongjin He.

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Jiang, F., Zhang, Z. & He, H. Solving saddle point problems: a landscape of primal-dual algorithm with larger stepsizes. J Glob Optim 85, 821–846 (2023). https://doi.org/10.1007/s10898-022-01233-0

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  • DOI: https://doi.org/10.1007/s10898-022-01233-0

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