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Iteration-Complexity of First-Order Augmented Lagrangian Methods for Convex Conic Programming

Zhaosong Lu (zhaosong***at***sfu.ca)
Zirui Zhou (zrzhou01***at***gmail.com)

Abstract: 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.

Keywords: Convex conic programming, augmented Lagrangian method, first-order method, iteration complexity

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Citation: Zhaosong Lu and Zirui Zhou. Iteration-Complexity of First-Order Augmented Lagrangian Methods for Convex Conic Programming. arXiv:1803.09941, preprint, 2018.

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Entry Submitted: 03/28/2018
Entry Accepted: 03/28/2018
Entry Last Modified: 03/29/2018

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