In this article we present a version of the proximal alternating direction method for a convex problem with linear constraints and a separable objective function, in which the standard quadratic regularizing term is replaced with an interior proximal metric for those variables that are required to satisfy some additional convex constraints. Moreover, the proposed method has the advantage that t...

Recently several, so-called, proximal Newton methods were proposed for sparse optimization [6, 11, 8, 3]. These methods construct a composite quadratic approximation using Hessian information, optimize this approximation using a first-order method, such as coordinate descent and employ a line search to ensure sufficient descent. Here we propose a general framework, which includes slightly modif...

We present algorithms for square classes, quadratic forms and Witt classes of quadratic forms over the eld of rational functions of one variable over the reals. The algorithms are capable of: nding the unique representative of a square class, deciding if a given function is a square or a sum of squares and deciding if a quadratic form is isotropic or hyperbolic. Moreover we propose a representa...

Two inexact versions of a Bregman-function-based proximal method for nding a zero of a maximal monotone operator, suggested in [4] and [3], are considered. For a wide class of Bregman functions convergence of these methods is proved under an essentially weaker condition on the sequence of errors than in the original papers. Also the error criterion of a logarithmic-quadratic proximal method, de...

This paper presents a new inexact proximal method for solving monotone variational inequality problems with a given separable structure. The resulting method combines the recent proximal distances theory introduced by Auslender and Teboulle (2006) with a decomposition method given by Chen and Teboulle that was proposed to solve convex optimization problems. This method extends and generalizes p...

An algorithm is developped for minimizing nonsmooth convex functions. This algortithm extends Elzinga-Moore cutting plane algorithm by enforcing the search of the next test point not too far from the previous ones, thus removing compactness assumption. Our method is to Elzinga-Moore’s algorithm what a proximal bundle method is to Kelley’s algorithm. As in proximal bundle methods, a quadratic pr...

In this paper, we proposed a logarithmic-quadratic proximal alternating direction method for structured variational inequalities. The new iterate is obtained by a convex combination of the previous point and the one generated by a projection-type method along a new descent direction. Global convergence of the new method is proved under certain assumptions. We also reported some numerical result...

The conjugate gradient (CG) method is an e cient iterative method for solving large-scale strongly convex quadratic programming (QP). In this paper we propose some generalized CG (GCG) methods for solving the `1-regularized (possibly not strongly) convex QP that terminate at an optimal solution in a nite number of iterations. At each iteration, our methods rst identify a face of an orthant and ...

This paper presents a new computational approach for solving the Regularized Total Least Squares problem. The problem is formulated by adding a quadratic constraint to the Total Least Square minimization problem. Starting from the fact that a quadratically constrained Least Squares problem can be solved via a quadratic eigenvalue problem, an iterative procedure for solving the regularized Total...

This paper presents a new and efficient method for video data compression using quadratic Bézier curve fitting. The method treats the luminance or color variations of a spatial location in a sequence of frames as input points in Euclidean space R1 or R3. The input points are approximated using quadratic Bézier least square fitting. The output data consists of quadratic Bézier control points and...