Second-order orthant-based methods with enriched Hessian information for sparse $$\ell _1$$ ℓ 1 -optimization

J. C. De Los Reyes (), E. Loayza and P. Merino
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J. C. De Los Reyes: Escuela Politécnica Nacional
E. Loayza: Escuela Politécnica Nacional
P. Merino: Escuela Politécnica Nacional

Computational Optimization and Applications, 2017, vol. 67, issue 2, No 1, 225-258

Abstract: Abstract We present a second order algorithm, based on orthantwise directions, for solving optimization problems involving the sparsity enhancing $$\ell _1$$ ℓ 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 $$\ell _1$$ ℓ 1 -norm. The weak second order information behind the $$\ell _1$$ ℓ 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.

Keywords: Sparse optimization; Orthantwise directions; Second-order algorithms; Semismooth Newton methods; 49M15; 65K05; 90C53; 49J20; 49K20 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10589-017-9891-z

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