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<p>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.</p>

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

  • 期刊论文

  • 发表日期：

    2025年1月16日

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  • Xiaoquan Wang,
  • Hu Shao,
  • Ting Wu

摘要

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.