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

Author

Listed:
  • Fan Jiang

    (Nanjing University of Information Science and Technology)

  • Zhiyuan Zhang

    (Xiamen University)

  • Hongjin He

    (Ningbo University)

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.

Suggested Citation

  • Fan Jiang & Zhiyuan Zhang & Hongjin He, 2023. "Solving saddle point problems: a landscape of primal-dual algorithm with larger stepsizes," Journal of Global Optimization, Springer, vol. 85(4), pages 821-846, April.
    • Handle: RePEc:spr:jglopt:v:85:y:2023:i:4:d:10.1007_s10898-022-01233-0
    DOI: 10.1007/s10898-022-01233-0
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    References listed on IDEAS

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    1. Hongjin He & Jitamitra Desai & Kai Wang, 2016. "A primal–dual prediction–correction algorithm for saddle point optimization," Journal of Global Optimization, Springer, vol. 66(3), pages 573-583, November.
    2. Xingju Cai & Deren Han & Lingling Xu, 2013. "An improved first-order primal-dual algorithm with a new correction step," Journal of Global Optimization, Springer, vol. 57(4), pages 1419-1428, December.
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