Convergence rate of inexact augmented Lagrangian method with practical relative error criterion for composite convex programming

In this paper, we consider the composite convex optimization problem with a linear equality constraint. We propose a practical inexact augmented Lagrangian (IAL) framework that employs two relative error criteria. Under the first criterion, we demonstrate convergence and establish sublinear ergodic convergence rates. By incorporating the second criterion, we achieve sublinear non-ergodic convergence rates. Furthermore, we determine the total iteration complexity of the IAL framework by slightly relaxing these criteria. Numerical experiments on both synthetic and real-world problems are conducted to illustrate the efficiency of the proposed IAL method.