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.
Smoothing Alternating Direction Methods for Fully Nonsmooth Constrained Convex Optimization
