In this paper we consider a class of convex conic programming. In particular, we propose an inexact augmented Lagrangian (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 computing an $\epsilon$-KKT solution is at most $\mathcal{O}(\epsilon^{-7/4})$. We also propose a modified I-AL method and show that it has an improved iteration-complexity $\mathcal{O}(\epsilon^{-1}\log\epsilon^{-1})$, which is so far the lowest complexity bound among all first-order I-AL type of methods for computing an $\epsilon$-KKT solution. Our complexity analysis of the I-AL methods is mainly based on an analysis on inexact proximal point algorithm (PPA) and the link between the I-AL methods and inexact PPA. It is substantially different from the 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. Compared to the mostly related I-AL methods \cite{Lan16}, our modified I-AL method is more practically efficient and also applicable to a broader class of problems.

中文翻译：

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凸圆锥规划的一阶增广拉格朗日方法的迭代复杂度

在本文中，我们考虑一类凸圆锥规划。特别是，我们提出了一种非精确增广拉格朗日 (I-AL) 方法来解决这个问题，其中增广拉格朗日子问题通过 Nesterov 最优一阶方法的变体近似求解。我们表明，用于计算 $\epsilon$-KKT 解决方案的 I-AL 方法的一阶迭代总数最多为 $\mathcal{O}(\epsilon^{-7/4})$。我们还提出了一种改进的 I-AL 方法，并表明它具有改进的迭代复杂度 $\mathcal{O}(\epsilon^{-1}\log\epsilon^{-1})$，这是迄今为止用于计算 $\epsilon$-KKT 解的所有一阶 I-AL 类型方法中的最低复杂度界限。我们对 I-AL 方法的复杂性分析主要基于对不精确近点算法 (PPA) 以及 I-AL 方法与不精确 PPA 之间的联系的分析。它与文献中现有的一阶 I-AL 方法的复杂性分析有很大不同，后者通常将 I-AL 方法视为不精确的双梯度方法。与最相关的 I-AL 方法 \cite{Lan16} 相比，我们改进的 I-AL 方法在实践中更高效，也适用于更广泛的问题。