We propose two classes of second-order optimization methods for solving the sparse inverse covariance estimation problem. The first approach, which we call the Newton-LASSO method, minimizes a piecewise quadratic model of the objective function at every iteration to generate a step. We employ the fast iterative shrinkage thresholding method (FISTA) to solve this subproblem. The … Read more
In this paper, we introduce a new stochastic approximation (SA) type algorithm, namely the randomized stochastic gradient (RSG) method, for solving an important class of nonlinear (possibly nonconvex) stochastic programming (SP) problems. We establish the complexity of this method for computing an approximate stationary point of a nonlinear programming problem. We also show that this … Read more
We study sparse approximate solutions to convex optimization problems. It is known that in many engineering applications researchers are interested in an approximate solution of an optimization problem as a linear combination of elements from a given system of elements. There is an increasing interest in building such sparse approximate solutions using different greedy-type algorithms. … Read more
This paper is a follow up to the previous author’s paper on convex optimization. In that paper we began the process of adjusting greedy-type algorithms from nonlinear approximation for finding sparse solutions of convex optimization problems. We modified there three the most popular in nonlinear approximation in Banach spaces greedy algorithms — Weak Chebyshev Greedy … Read more
We present a homogeneous algorithm equipped with a modified potential function for the monotone complementarity problem. We show that this potential function is reduced by at least a constant amount if a scaled Lipschitz condition is satis ed. A practical algorithm based on this potential function is implemented in a software package named iOptimize. The implementation … Read more
In this paper we present a variant of the conjugate gradient (CG) algorithm in which we invoke a subspace minimization subproblem on each iteration. We call this algorithm CGSO for “conjugate gradient with subspace optimization”. It is related to earlier work by Nemirovsky and Yudin. We apply the algorithm to solve unconstrained strictly convex problems. … Read more
We consider a new class of huge-scale problems, the problems with {\em sparse subgradients}. The most important functions of this type are piece-wise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient iterations, which total cost depends {\em logarithmically} in the dimension. This technique is … Read more
This article addresses a general criterion providing a zero duality gap for convex programs in the setting of the real locally convex spaces. The main theorem of our work is formulated only in terms of the constraints of the program, hence it holds true for any objective function fulfilling a very general qualification condition, implied … Read more
This paper takes a uniform look at the customized applications of proximal point algorithm (PPA) to two classes of problems: the linearly constrained convex minimization problem with a generic or separable objective function and a saddle-point problem. We model these two classes of problems uniformly by a mixed variational inequality, and show how PPA with … Read more
We consider a general class of convex optimization problems in which one seeks to minimize a strongly convex function over a closed and convex set which is by itself an optimal set of another convex problem. We introduce a gradient-based method, called the minimal norm gradient method, for solving this class of problems, and establish … Read more