# American Institute of Mathematical Sciences

2013, 3(2): 247-260. doi: 10.3934/naco.2013.3.247

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

 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  April 2013

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.
Citation: Bingsheng He, Xiaoming Yuan. Linearized alternating direction method of multipliers with Gaussian back substitution for separable convex programming. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 247-260. doi: 10.3934/naco.2013.3.247
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