  In this paper we propose a second--order method for solving \emph{linear
composite sparse optimization problems} consisting of minimizing the sum of a
differentiable (possibly nonconvex function) and a nondifferentiable convex
term. The composite nondifferentiable convex penalizer is given by
$\ell_1$--norm of a matrix multiplied with the coefficient vector. The
algorithm that we propose for the case of the linear composite $\ell_1$ problem
relies on the three main ingredients that power the OESOM algorithm
\cite{dlrlm07}: the minimum norm subgradient, a projection step and, in
particular, the second--order information associated to the nondifferentiable
term. By extending these devices, we obtain a full second--order method for
solving composite sparse optimization problems which includes a wide range of
applications. For instance, problems involving the minimization of a general
class \emph{differential graph operators} can be solved with the proposed
algorithm. We present several computational experiments to show the efficiency
of our approach for different application examples.
