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 (AL) 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 non-ergodic convergence rate of the proposed IAL framework.
Ya-Feng Liu, Xin Liu, and Shiqian Ma, "On the non-ergodic convergence rate of an inexact augmented Lagrangian framework for composite convex programming," Academy of Mathematics and Systems Science, Chinese Academy of Sciences, October, 2017.