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.

翻译：在本文中,我们考虑的是一组同族同族同族同族同族同族的编程。特别是,我们首先建议一种不确切的扩大拉格朗吉亚(I-AL)方法,该方法类似于传统的I-AL方法,用以解决这一问题。在这个方法中,增加拉格朗吉亚子问题基本上通过Nesterov最佳第一阶方法的变体来解决。我们表明,提议的I-AL方法的一级迭代总数最多为$\messilon$-KKT 方法的美元。我们随后建议一种适应性化的标准化I-AL方法,并表明该方法实现了一阶迭代复杂程度的复杂性$\mathcal{O}(\ exloná- 1\ log\ epsilon_1}} 美元。我们显示,提议的I-AL方法的第一阶I-AL方法的目前复杂程度,即I-al-lassy 方法的双级同族同族之间,我们I-AL方法的复杂程度分析基于对I-exact AL方法的精确分析。