In this paper we consider sparse approximation problems, that is, general $l_0$ minimization problems with the $l_0$-“norm” of a vector being a part of constraints or objective function. In particular, we first study the first-order optimality conditions for these problems. We then propose penalty decomposition (PD) methods for solving them in which a sequence of … Read more
In this paper we consider general l0-norm minimization problems, that is, the problems with l0-norm appearing in either objective function or constraint. In particular, we first reformulate the l0-norm constrained problem as an equivalent rank minimization problem and then apply the penalty decomposition (PD) method proposed in [33] to solve the latter problem. By utilizing … Read more
Sparse covariance selection problems can be formulated as log-determinant (log-det) semidefinite programming (SDP) problems with large numbers of linear constraints. Standard primal-dual interior-point methods that are based on solving the Schur complement equation would encounter severe computational bottlenecks if they are applied to solve these SDPs. In this paper, we consider a customized inexact primal-dual … Read more
We propose a Newton-CG primal proximal point algorithm for solving large scale log-determinant optimization problems. Our algorithm employs the essential ideas of the proximal point algorithm, the Newton method and the preconditioned conjugate gradient solver. When applying the Newton method to solve the inner sub-problem, we find that the log-determinant term plays the role of … Read more
In this paper, we consider estimating sparse inverse covariance of a Gaussian graphical model whose conditional independence is assumed to be partially known. Similarly as in [5], we formulate it as an $l_1$-norm penalized maximum likelihood estimation problem. Further, we propose an algorithm framework, and develop two first-order methods, that is, adaptive spectral projected gradient … Read more