On the Nonergodic Convergence Rate of an Inexact Augmented Lagrangian Framework for Composite Convex Programming

• 成果类型: Article
• 署名作者: Liu, Ya-Feng; Liu, Xin; Ma, Shiqian
• 署名单位: Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS; Chinese Academy of Sciences; Academy of Mathematics & System Sciences, CAS; Chinese Academy of Sciences; University of Chinese Academy of Sciences, CAS; University of California System; University of California Davis
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2018.0939
• 发表日期: 2019
• 页码: 632-650
• 关键词: gradient methods; complexity

摘要

In this paper, we consider the linearly constrained composite convex optimization problem, whose objective is a sum of a smooth function and a possibly nonsmooth function. We propose an inexact augmented Lagrangian (IAL) framework for solving the problem. The stopping criterion used in solving the augmented Lagrangian subproblem in the proposed IAL framework is weaker and potentially much easier to check than the one used in most of the existing IAL frameworks/methods. We analyze the global convergence and the nonergodic convergence rate of the proposed IAL framework. Preliminary numerical results are presented to show the efficiency of the proposed IAL framework and the importance of the nonergodic convergence and convergence rate analysis.