Document Type

Journal Article

Department/Unit

Department of Mathematics

Title

Alternating direction method with Gaussian back substitution for separable convex programming

Language

English

Abstract

We consider the linearly constrained separable convex minimization problem whose objective function is separable into m individual convex functions with nonoverlapping variables. A Douglas-Rachford alternating direction method of multipliers (ADM) has been well studied in the literature for the special case of m = 2. But the convergence of extending ADM to the general case of m ≥ 3 is still open. In this paper, we show that the straightforward extension of ADM is valid for the general case of m ≥ 3 if it is combined with a Gaussian back substitution procedure. The resulting ADM with Gaussian back substitution is a novel approach towards the extension of ADM from m = 2 to m ≥ 3, and its algorithmic framework is new in the literature. For the ADM with Gaussian back substitution, we prove its convergence via the analytic framework of contractive-type methods, and we show its numerical efficiency by some application problems. © 2012 Society for Industrial and Applied Mathematics.

Keywords

Alternating direction method, Convex programming, Gaussian back substitution, Separable structure

Publication Date

2012

Source Publication Title

SIAM Journal on Optimization

Volume

22

Issue

2

Start Page

313

End Page

340

Publisher

Society for Industrial and Applied Mathematics

DOI

10.1137/110822347

Link to Publisher's Edition

http://dx.doi.org/10.1137/110822347

ISSN (print)

10526234

ISSN (electronic)

10957189

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