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

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. Funding: The work of Y.-F. Liu was supported in part by National Natural Science Foundation of China (NSFC) [Grants 11631013, 11331012, 11671419, and 11571221] and Beijing Natural Science Foundation [Grant LI72020]. The work of X. Liu was supported in part by NSFC [Grants 11622112, 11471325, 91530204, and 11688101], the National Center for Mathematics and Interdisciplinary Sciences, Chinese Academy of Sciences (CAS), and Key Research Program of Frontier Sciences (QYZDJ-SSW-SYS010), CAS. The work of S. Ma was supported in part by a startup package from the Department of Mathematics at University of California, Davis. Keywords: inexact augmented Lagrangian framework * nonergodic convergence rate * composite convex programming
1. Introduction In this paper, we consider the linearly constrained composite convex optimization problem [mathematical expression not reproducible] (1) where A [member of] [R.sup.mxn] and b [member of] [R.sup.m]; f(x) [...]