Please use this identifier to cite or link to this item: https://doi.org/10.1007/s10107-011-0471-1
Title: A block coordinate gradient descent method for regularized convex separable optimization and covariance selection
Authors: Yun, S.
Tseng, P.
Toh, K.-C. 
Keywords: ℓ1- penalization
Block coordinate gradient descent
Complexity
Convex optimization
Covariance selection
Global convergence
Linear rate convergence
Maximum likelihood estimation
Issue Date: Oct-2011
Citation: Yun, S., Tseng, P., Toh, K.-C. (2011-10). A block coordinate gradient descent method for regularized convex separable optimization and covariance selection. Mathematical Programming 129 (2) : 331-355. ScholarBank@NUS Repository. https://doi.org/10.1007/s10107-011-0471-1
Abstract: We consider a class of unconstrained nonsmooth convex optimization problems, in which the objective function is the sum of a convex smooth function on an open subset of matrices and a separable convex function on a set of matrices. This problem includes the covariance selection problem that can be expressed as an ℓ1-penalized maximum likelihood estimation problem. In this paper, we propose a block coordinate gradient descent method (abbreviated asBCGD)for solving this class of nonsmooth separable problems with the coordinate block chosen by a Gauss-Seidel rule. The method is simple, highly parallelizable, and suited for large-scale problems. We establish global convergence and, under a local Lipschizian error bound assumption, linear rate of convergence for this method. For the covariance selection problem, the method can terminate in O(n3/ε) iterations with an ε-optimal solution. We compare the performance of the BCGD method with the first-order methods proposed by Lu (SIAM J Optim 19:1807-1827, 2009; SIAM J Matrix Anal Appl 31:2000-2016, 2010) for solving the covariance selection problem on randomly generated instances. Our numerical experience suggests that the BCGD method can be efficient for largescale covariance selection problems with constraints. © Springer and Mathematical Optimization Society 2011.
Source Title: Mathematical Programming
URI: http://scholarbank.nus.edu.sg/handle/10635/102605
ISSN: 00255610
DOI: 10.1007/s10107-011-0471-1
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