Second-order orthant-based methods with enriched Hessian information for sparse $$\ell _1$$-optimization |
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Authors: | J C De Los Reyes E Loayza P Merino |
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Institution: | 1.Research Center on Mathematical Modeling (MODEMAT),Escuela Politécnica Nacional,Quito,Ecuador |
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Abstract: | We present a second order algorithm, based on orthantwise directions, for solving optimization problems involving the sparsity enhancing \(\ell _1\)-norm. The main idea of our method consists in modifying the descent orthantwise directions by using second order information both of the regular term and (in weak 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 the main features of our algorithm consists in a faster identification of the active set. We also prove that a reduced version of our method is equivalent to a semismooth Newton algorithm applied to the optimality condition, under a specific choice of the algorithm parameters. We present several computational experiments to show the efficiency of our approach compared to other state-of-the-art algorithms. |
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