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An enriched second-order method for nonconvex composite sparse optimization problems

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.