Iteration complexity of first-order augmented Lagrangian methods for convex conic programming

In this paper we consider a class of convex conic programming. In particular, we first propose an inexact augmented Lagrangian (I-AL) method that resembles the classical I-AL method for solving this problem, in which the augmented Lagrangian subproblems are solved approximately by a variant of Nesterov's optimal first-order method. We show that the total number of first-order iterations of the proposed I-AL method for finding an ε-KKT solution is at most O(ε-7/4). We then propose an adaptively regularized I-AL method and show that it achieves a first-order iteration complexity O(ε-1logε-1), which significantly improves existing complexity bounds achieved by first-order I-AL methods for finding an ε-KKT solution. Our complexity analysis of the I-AL methods is based on a sharp analysis of inexact proximal point algorithm (PPA) and the connection between the I-AL methods and inexact PPA. It is vastly different from existing complexity analyses of the first-order I-AL methods in the literature, which typically regard the I-AL methods as an inexact dual gradient method.