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We present a second order algorithm, based on orthantwise directions, forsolving optimization problems involving the sparsity enhancing $\ell_1$-norm.The main idea of our method consists in modifying the descent orthantwisedirections by using second order information both of the regular term and (inweak sense) of the $\ell_1$-norm. The weak second order information behind the$\ell_1$-term is incorporated via a partial Huber regularization. One of themain features of our algorithm consists in a faster identification of theactive set. We also prove that a reduced version of our method is equivalent toa semismooth Newton algorithm applied to the optimality condition, under aspecific choice of the algorithm parameters. We present several computationalexperiments to show the efficiency of our approach compared to otherstate-of-the-art algorithms.

Second-order orthant-based methods with enriched Hessian information for sparse $\ell_1$-optimization