scholarly journals Infeasibility Detection in the Alternating Direction Method of Multipliers for Convex Optimization

2018 ◽  
Author(s):  
Goran Banjac ◽  
Paul Goulart ◽  
Bartolomeo Stellato ◽  
Stephen Boyd
Keyword(s):  
Convex Optimization ◽  
Alternating Direction Method ◽  
Method Of Multipliers ◽  
Alternating Direction ◽  
Infeasibility Detection

scholarly journals Alternating Direction Methods for Latent Variable Gaussian Graphical Model Selection

2013 ◽  
Vol 25 (8) ◽  
pp. 2172-2198 ◽  
Author(s):  
Shiqian Ma ◽  
Lingzhou Xue ◽  
Hui Zou
Keyword(s):  
Model Selection ◽  
Convex Optimization ◽  
Optimization Problem ◽  
Graphical Model ◽  
Sparse Matrix ◽  
Alternating Direction Method ◽  
Method Of Multipliers ◽  
Convex Optimization Problem ◽  
Alternating Direction Methods ◽  
Alternating Direction

Chandrasekaran, Parrilo, and Willsky ( 2012 ) proposed a convex optimization problem for graphical model selection in the presence of unobserved variables. This convex optimization problem aims to estimate an inverse covariance matrix that can be decomposed into a sparse matrix minus a low-rank matrix from sample data. Solving this convex optimization problem is very challenging, especially for large problems. In this letter, we propose two alternating direction methods for solving this problem. The first method is to apply the classic alternating direction method of multipliers to solve the problem as a consensus problem. The second method is a proximal gradient-based alternating-direction method of multipliers. Our methods take advantage of the special structure of the problem and thus can solve large problems very efficiently. A global convergence result is established for the proposed methods. Numerical results on both synthetic data and gene expression data show that our methods usually solve problems with 1 million variables in 1 to 2 minutes and are usually 5 to 35 times faster than a state-of-the-art Newton-CG proximal point algorithm.

scholarly journals Alternating Direction Method of Multipliers for Separable Convex Optimization of Real Functions in Complex Variables

2015 ◽  
Vol 2015 ◽  
pp. 1-14 ◽  
Author(s):  
Lu Li ◽  
Xingyu Wang ◽  
Guoqiang Wang
Keyword(s):  
Convex Optimization ◽  
Complex Variables ◽  
Alternating Direction Method ◽  
Projection Algorithm ◽  
Real Field ◽  
Method Of Multipliers ◽  
The Real ◽  
Alternating Direction ◽  
Separable Convex Optimization ◽  
Real Functions

The alternating direction method of multipliers (ADMM) has been widely explored due to its broad applications, and its convergence has been gotten in the real field. In this paper, an ADMM is presented for separable convex optimization of real functions in complex variables. First, the convergence of the proposed method in the complex domain is established by using the Wirtinger Calculus technique. Second, the basis pursuit (BP) algorithm is given in the form of ADMM in which the projection algorithm and the soft thresholding formula are generalized from the real case. The numerical simulations on the reconstruction of electroencephalogram (EEG) signal are provided to show that our new ADMM has better behavior than the classic ADMM for solving separable convex optimization of real functions in complex variables.

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