Zhaosong Lu In this paper we consider $l_0$ regularized convex cone programming problems. In particular, we first propose an iterative hard thresholding (IHT) method and its variant for solving $l_0$ regularized box constrained convex programming. We show that the sequence generated by these methods converges to a local minimizer. Also, we establish the iteration complexity of the … Read more
In this paper we study general $l_p$ regularized unconstrained minimization problems. In particular, we derive lower bounds for nonzero entries of first- and second-order stationary points, and hence also of local minimizers of the $l_p$ minimization problems. We extend some existing iterative reweighted $l_1$ (IRL1) and $l_2$ (IRL2) minimization methods to solve these problems and … Read more
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 propose a primal IP method for solving the optimal experimental design problem with a large class of smooth convex optimality criteria, including A-, D- and p th mean criterion, and establish its global convergence. We also show that the Newton direction can be computed efficiently when the size of the moment … Read more
In this paper we study interior point (IP) methods for solving optimal design problems. In particular, we propose a primal IP method for solving the problems with general convex optimality criteria and establish its global convergence. In addition, we reformulate the problems with A-, D- and E-criterion into linear or log-determinant semidefinite programs (SDPs) and … Read more
In this paper we consider general rank minimization problems with rank appearing in either objective function or constraint. We first show that a class of matrix optimization problems can be solved as lower dimensional vector optimization problems. As a consequence, we establish that a class of rank minimization problems have closed form solutions. Using this … 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
Principal component analysis (PCA) is a widely used technique for data analysis and dimension reduction with numerous applications in science and engineering. However, the standard PCA suffers from the fact that the principal components (PCs) are usually linear combinations of all the original variables, and it is thus often difficult to interpret the PCs. To … 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
In this paper, we study a gradient based method for general cone programming (CP) problems. In particular, we first consider four natural primal-dual convex smooth minimization reformulations for them, and then discuss a variant of Nesterov’s smooth (VNS) method recently proposed by Tseng [30] for solving these reformulations. The associated worst-case major arithmetic operations costs … Read more