  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 $\epsilon$-KKT solution
is at most $\mathcal{O}(\epsilon^{-7/4})$. We then propose an adaptively
regularized I-AL method and show that it achieves a first-order iteration
complexity $\mathcal{O}(\epsilon^{-1}\log\epsilon^{-1})$, which significantly
improves existing complexity bounds achieved by first-order I-AL methods for
finding an $\epsilon$-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.
