The exact worst-case convergence rate of the alternating direction method of multipliers

• 成果类型: Article
• 署名作者: Zamani, Moslem; Abbaszadehpeivasti, Hadi; de Klerk, Etienne
• 署名单位: Tilburg University; Universite Catholique Louvain
• 刊物名称: MATHEMATICAL PROGRAMMING
• ISSN/ISSBN: 0025-5610
• DOI: 10.1007/s10107-023-02037-0
• 发表日期: 2024
• 页码: 243-276
• 关键词: 1st-order methods; performance; optimization; complexity; rachford

摘要

Recently, semidefinite programming performance estimation has been employed as a strong tool for the worst-case performance analysis of first order methods. In this paper, we derive new non-ergodic convergence rates for the alternating direction method of multipliers (ADMM) by using performance estimation. We give some examples which show the exactness of the given bounds. We also study the linear and R-linear convergence of ADMM in terms of dual objective. We establish that ADMM enjoys a global linear convergence rate if and only if the dual objective satisfies the Polyak-Lojasiewicz (PL) inequality in the presence of strong convexity. In addition, we give an explicit formula for the linear convergence rate factor. Moreover, we study the R-linear convergence of ADMM under two scenarios.