Optimization over Sparse Symmetric Sets via a Nonmonotone Projected Gradient Method

We consider the problem of minimizing a Lipschitz differentiable function over a class of sparse symmetric sets that has wide applications in engineering and science. For this problem, it is known that any accumulation point of the classical projected gradient (PG) method with a constant stepsize $1/L$ satisfies the $L$-stationarity optimality condition that was introduced … Read more

A proximal gradient method for ensemble density functional theory

The ensemble density functional theory is valuable for simulations of metallic systems due to the absence of a gap in the spectrum of the Hamiltonian matrices. Although the widely used self-consistent field iteration method can be extended to solve the minimization of the total energy functional with respect to orthogonality constraints, there is no theoretical … Read more

An Accelerated Proximal Coordinate Gradient Method and its Application to Regularized Empirical Risk Minimization

We consider the problem of minimizing the sum of two convex functions: one is smooth and given by a gradient oracle, and the other is separable over blocks of coordinates and has a simple known structure over each block. We develop an accelerated randomized proximal coordinate gradient (APCG) method for minimizing such convex composite functions. … Read more

Randomized Block Coordinate Non-Monotone Gradient Method for a Class of Nonlinear Programming

In this paper we propose a randomized block coordinate non-monotone gradient (RBCNMG) method for minimizing the sum of a smooth (possibly nonconvex) function and a block-separable (possibly nonconvex nonsmooth) function. At each iteration, this method randomly picks a block according to any prescribed probability distribution and typically solves several associated proximal subproblems that usually have … Read more

On the Complexity Analysis of Randomized Block-Coordinate Descent Methods

In this paper we analyze the randomized block-coordinate descent (RBCD) methods for minimizing the sum of a smooth convex function and a block-separable convex function. In particular, we extend Nesterov’s technique (SIOPT 2012) for analyzing the RBCD method for minimizing a smooth convex function over a block-separable closed convex set to the aforementioned more general … Read more

Iterative Hard Thresholding Methods for $ Regularized Convex Cone Programming

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

Iterative Reweighted Minimization Methods for $ Regularized Unconstrained Nonlinear Programming

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

Sparse Approximation via Penalty Decomposition Methods

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

Interior Point Methods for Optimal Experimental Designs

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

Interior Point Methods for Computing Optimal Designs

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