Alternating Direction Methods for Sparse Covariance Selection

The mathematical model of the widely-used sparse covariance selection problem (SCSP) is an NP-hard combinatorial problem, whereas it can be well approximately by a convex relaxation problem whose maximum likelihood estimation is penalized by the $L_1$ norm. This convex relaxation problem, however, is still numerically challenging, especially for large-scale cases. Recently, some efficient first-order methods … Read more

A First-Order Smoothed Penalty Method for Compressed Sensing

We propose a first-order smoothed penalty algorithm (SPA) to solve the sparse recovery problem min{||x||_1 : Ax=b}. SPA is efficient as long as the matrix-vector product Ax and A^Ty can be computed efficiently; in particular, A need not be an orthogonal projection matrix. SPA converges to the target signal by solving a sequence of penalized … Read more

Iteration-complexity of first-order augmented Lagrangian methods for convex programming

This paper considers a special class of convex programming (CP) problems whose feasible regions consist of a simple compact convex set intersected with an affine manifold. We present first-order methods for this class of problems based on an inexact version of the classical augmented Lagrangian (AL) approach, where the subproblems are approximately solved by means … Read more

Convex Optimization Methods for Dimension Reduction and Coefficient Estimation in Multivariate Linear Regression

In this paper, we study convex optimization methods for computing the trace norm regularized least squares estimate in multivariate linear regression. The so-called factor estimation and selection (FES) method, recently proposed by Yuan et al. [17], conducts parameter estimation and factor selection simultaneously and have been shown to enjoy nice properties in both large and … Read more

Sparse Covariance Selection via Robust Maximum Likelihood Estimation

We address a problem of covariance selection, where we seek a trade-off between a high likelihood against the number of non-zero elements in the inverse covariance matrix. We solve a maximum likelihood problem with a penalty term given by the sum of absolute values of the elements of the inverse covariance matrix, and allow for … Read more

First- and Second-Order Methods for Semidefinite Programming

In this paper, we survey the most recent methods that have been developed for the solution of semidefinite programs. We first concentrate on the methods that have been primarily motivated by the interior point (IP) algorithms for linear programming, putting special emphasis in the class of primal-dual path-following algorithms. We also survey methods that have … Read more