An Inexact Linearized Augmented Lagrangian Method for the Linearly Composite Convex Programming

We propose an inexact linearized augmented Lagrangian method for the linear equality constrained convex programming, for which the objective function has a “nonsmooth + smooth” composite structure. We show that both the objective error and the constraint gap associated with the proposed algorithm enjoy an ${𝒪}(\frac{1}{k})$ nonergodic convergence rate. By choosing a specific proximal matrix, we drive a customized linearized augmented Lagrangian method for the problem, where the subproblem can be solved by means of the proximal mapping of a convex function. Finally, we demonstrate the efficiency of the proposed algorithms by two numerical experiments.