Work

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

Year: 2023

Type: article

Abstract: . 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 [Formula: see text]-KKT solution is at most [Formula: see text]. We then propose an adaptively regularized I-AL method and show that it achieves a first-order iteration complexity [Formula: see text], which significantly improves existing complexity bounds achieved by first-order I-AL methods for finding an [Formula: see text]-KKT solution. Our complexity analysis of the I-AL methods is based on a sharp analysis of the 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.

Source: SIAM Journal on Optimization

DOI: 10.1137/21m1403837

Authors: Zhaosong Lu, Zirui Zhou

Institutions: University of Minnesota, Huawei Technologies (Canada)

Language: English

Topic: Optimization and Variational Analysis

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, National Natural Science Foundation of China

Awards: G1695691088, Collaborative Research: III: Medium: New Machine Learning Empowered Nanoinformatics System for Advancing Nanomaterial Design