## A proximal minimization algorithm for structured nonconvex and nonsmooth problems

We propose a proximal algorithm for minimizing objective functions consisting of three summands: the composition of a nonsmooth function with a linear operator, another nonsmooth function, each of the nonsmooth summands depending on an independent block variable, and a smooth function which couples the two block variables. The algorithm is a full splitting method, which … Read more

In this paper we propose an adaptively extrapolated proximal gradient method, which is based on the accelerated proximal gradient method (also known as FISTA), however we locally optimize the extrapolation parameter by carrying out an exact (or inexact) line search. It turns out that in some situations, the proposed algorithm is equivalent to a class … Read more

## New Improved Penalty Methods for Sparse Reconstruction Based on Difference of Two Norms

In this paper, we further establish two types of DC (Difference of Convex functions) programming for $l_0$ sparse reconstruction. Our DC objective functions are specified to the difference of two norms. One is the difference of $l_1$ and $l_{\sigma_q}$ norms (DC $l_1$-$l_{\sigma_q}$ for short) where $l_{\sigma_q}$ is the $l_1$ norm of the $q$-term ($q\geq1$) best … Read more

## Alternating proximal algorithms for constrained variational inequalities. Application to domain decomposition for PDE’s

Let $\cX,\cY,\cZ$ be real Hilbert spaces, let $f : \cX \rightarrow \R\cup\{+\infty\}$, $g : \cY \rightarrow \R\cup\{+\infty\}$ be closed convex functions and let $A : \cX \rightarrow \cZ$, $B : \cY \rightarrow \cZ$ be linear continuous operators. Let us consider the constrained minimization problem $$\min\{f(x)+g(y):\quad Ax=By\}.\leqno (\cP)$$ Given a sequence $(\gamma_n)$ which tends toward … Read more

## Global linear convergence of an augmented Lagrangian algorithm for solving convex quadratic optimization problems

We consider an augmented Lagrangian algorithm for minimizing a convex quadratic function subject to linear inequality constraints. Linear optimization is an important particular instance of this problem. We show that, provided the augmentation parameter is large enough, the constraint value converges {\em globally\/} linearly to zero. This property is viewed as a consequence of the … Read more