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A convex combined symmetric alternating direction method of multipliers for separable optimization

Author

Listed:
  • Xiaoquan Wang

    (China University of Mining and Technology)

  • Hu Shao

    (China University of Mining and Technology)

  • Ting Wu

    (Nanjing University)

Abstract

The Alternating Direction Method of Multipliers (ADMM) is a powerful first-order method used in many practical separable optimization problems. In this paper, we propose a new variant of the symmetric ADMM, called the Convex Combined Symmetric ADMM (CcS-ADMM), by integrating a convex combination technique. CcS-ADMM retains all the favorable features of ADMM, including the ability to take full advantage of problem structures and global convergence under relaxed parameter conditions. Furthermore, using the moderate assumptions and primal-dual gap, we analyze the convergence and the O(1/N) ergodic convergence rate of the algorithm with convex setting. Additionally, we propose the convergence of the CcS-ADMM with nonconvex setting in Euclidean space under so called Kurdyka–Lojasiewicz property and some widely used assumptions, and we establish the pointwise iteration-complexity of CcS-ADMM with respect to the augmented Lagrangian function and the primal-dual residuals. Finally, we present the results from preliminary numerical experiments to demonstrate the performance of the proposed algorithms.

Suggested Citation

  • Xiaoquan Wang & Hu Shao & Ting Wu, 2025. "A convex combined symmetric alternating direction method of multipliers for separable optimization," Computational Optimization and Applications, Springer, vol. 90(3), pages 839-880, April.
    • Handle: RePEc:spr:coopap:v:90:y:2025:i:3:d:10.1007_s10589-025-00647-2
    DOI: 10.1007/s10589-025-00647-2
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    References listed on IDEAS

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    1. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
    2. Kristian Bredies & Hongpeng Sun, 2017. "A Proximal Point Analysis of the Preconditioned Alternating Direction Method of Multipliers," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 878-907, June.
    3. Bin Gao & Feng Ma, 2018. "Symmetric Alternating Direction Method with Indefinite Proximal Regularization for Linearly Constrained Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 176(1), pages 178-204, January.
    4. Miantao Chao & Yongxin Zhao & Dongying Liang, 2020. "A Proximal Alternating Direction Method of Multipliers with a Substitution Procedure," Mathematical Problems in Engineering, Hindawi, vol. 2020, pages 1-12, April.
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