Document Type

Journal Article

Department/Unit

Department of Mathematics

Title

Linearized augmented Lagrangian and alternating direction methods for nuclear norm minimization

Language

English

Abstract

The nuclear norm is widely used to induce low-rank solutions for many optimization problems with matrix variables. Recently, it has been shown that the augmented Lagrangian method (ALM) and the alternating direction method (ADM) are very efficient for many convex programming problems arising from various applications, provided that the resulting subproblems are sufficiently simple to have closed-form solutions. In this paper, we are interested in the application of the ALM and the ADM for some nuclear norm involved minimization problems. When the resulting subproblems do not have closed-form solutions, we propose to linearize these subproblems such that closed-form solutions of these linearized subproblems can be easily derived. Global convergence results of these linearized ALM and ADM are established under standard assumptions. Finally, we verify the effectiveness and efficiency of these new methods by some numerical experiments. © 2012 American Mathematical Society.

Keywords

Alternating direction method, Augmented Lagrangian method, Convex programming, Linearized, Low-rank, Nuclear norm

Publication Date

2013

Source Publication Title

Mathematics of Computation

Volume

82

Issue

281

Start Page

301

End Page

329

Publisher

American Mathematical Society

DOI

10.1090/S0025-5718-2012-02598-1

Link to Publisher's Edition

http://dx.doi.org/10.1090/S0025-5718-2012-02598-1

ISSN (print)

00255718

ISSN (electronic)

10886842

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