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中国工业与应用数学学会会刊
主管：中华人民共和国教育部
主办：西安交通大学
ISSN 1005-3085  CN 61-1269/O1
工程数学学报

工程数学学报 ›› 2025, Vol. 42 ›› Issue (4): 721-735.doi: 10.3969/j.issn.1005-3085.2025.04.010cstr: 32411.14.cjem.CN61-1269/O1.2025.04.010

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非凸一致性问题邻近对称ADMM的收敛性分析

张静雯1,   党亚峥1,   倪诗皓1,   乔俊伟2   

  1. 1. 上海理工大学管理学院，上海  200093

    2. 上海出版印刷高等专科学校，上海 200093

  • 收稿日期:2022-12-04 接受日期:2023-07-30 出版日期:2025-08-15 发布日期:2025-10-15
  • 通讯作者: 党亚峥 E-mail: jgdyz@163.com
  • 基金资助:
    上海高校特聘教授(东方学者)岗位计划资助 (TP2022126).

Convergence Analysis of Proximal Symmetric ADMM for Nonconvex Consensus Problem

ZHANG Jingwen1,   DANG Yazheng1,   NI Shihao1,   QIAO Junwei2   

  1. 1. Business School, University of Shanghai for Science & Technology, Shanghai 200093

    2. Shanghai Publishing and Printing College, Shanghai 200093
  • Received:2022-12-04 Accepted:2023-07-30 Online:2025-08-15 Published:2025-10-15
  • Contact: Y. Dang. E-mail address: jgdyz@163.com
  • Supported by:
    The Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning (TP2022126).

摘要：

交替方向乘子法 (Alternating Direction Method of Multipliers, ADMM) 求解两分块优化的研究已经逐渐完善，但对于非凸多分块优化的研究较少，提出了一种带松弛步长参数的对称邻近ADMM用于求解非凸一致性问题。在适当的假设条件下，证明了算法的全局收敛性。其次，在效益函数满足Kurdyka-{\L}ojasiewicz (KL) 性质时，证明了算法的强收敛性。最后，数值实验验证了算法的有效性。

关键词: 非凸优化, 一致性问题, 交替方向乘子法, Kurdyka-{\L}ojasiewicz性质, 收敛性

Abstract:

The researches on the alternating direction method of multipliers (ADMM) for solving two-block optimization have been gradually perfect. However, the studies on ADMM for solving nonconvex multi-block optimization are relatively few. In this paper, we propose a symmetric proximal ADMM with relaxation stepsize parameter for nonconvex multi-block optimization. Under some suitable conditions, the global convergence of the algorithm is established. Subsequently, the strong convergence of the algorithm is established when the benefit function satisfies the Kurdyka-{\L}ojasiewicz (KL) property. Finally, numerical experiments verify the effectiveness of the proposed method.

Key words: nonconvex optimization, consensus problem, alternating direction method of multipliers, Kurdyka-{\L}ojasiewicz property, convergence

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