Linearized alternating direction method of multipliers with Gaussianback substitution  for separable convex programming

Bingsheng He; Xiaoming Yuan

doi:10.3934/naco.2013.3.247

Article Contents

2013, Volume 3, Issue 2: 247-260. Doi: 10.3934/naco.2013.3.247

This issue Previous Article A unified maximum entropy method via spline functions for Frobenius-Perron operators Next Article The stationary iterations revisited

Linearized alternating direction method of multipliers with Gaussian back substitution for separable convex programming

Bingsheng He^1, and
Xiaoming Yuan^2,

1.
Department of Mathematics, Nanjing University, Nanjing, 210093
2.
Department of Mathematics, Hong Kong Baptist University, Hong Kong

Received: February 2012

Revised: January 2013

Published: March 2013

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Abstract

Recently, we have proposed combining the alternating direction method of multipliers (ADMM) with a Gaussian back substitution procedure for solving the convex minimization model with linear constraints and a general separable objective function, i.e., the objective function is the sum of many functions without coupled variables. In this paper, we further study this topic and show that the decomposed subproblems in the ADMM procedure can be substantially alleviated by linearizing the involved quadratic terms arising from the augmented Lagrangian penalty. When the resolvent operators of the separable functions in the objective have closed-form representations, embedding the linearization into the ADMM subproblems becomes necessary to yield easy subproblems with closed-form solutions. We thus show theoretically that the blend of ADMM, Gaussian back substitution and linearization works effectively for the separable convex minimization model under consideration.

Keywords:

Alternating direction method of multipliers,
Gaussian back substitution,
linearization,
resolvent operator,
separable convex programming.

Mathematics Subject Classification: Primary: 90C25, 65K05; Secondary: 94A08.

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