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

Author

Listed:
  • J. C. De Los Reyes

    (Escuela Politécnica Nacional)

  • E. Loayza

    (Escuela Politécnica Nacional)

  • P. Merino

    (Escuela Politécnica Nacional)

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.

Suggested Citation

  • J. C. De Los Reyes & E. Loayza & P. Merino, 2017. "Second-order orthant-based methods with enriched Hessian information for sparse $$\ell _1$$ ℓ 1 -optimization," Computational Optimization and Applications, Springer, vol. 67(2), pages 225-258, June.
    • Handle: RePEc:spr:coopap:v:67:y:2017:i:2:d:10.1007_s10589-017-9891-z
    DOI: 10.1007/s10589-017-9891-z
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Georg Stadler, 2009. "Elliptic optimal control problems with L 1 -control cost and applications for the placement of control devices," Computational Optimization and Applications, Springer, vol. 44(2), pages 159-181, November.
    3. Emilie Chouzenoux & Jean-Christophe Pesquet & Audrey Repetti, 2014. "Variable Metric Forward–Backward Algorithm for Minimizing the Sum of a Differentiable Function and a Convex Function," Journal of Optimization Theory and Applications, Springer, vol. 162(1), pages 107-132, July.
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    Cited by:

    1. Pedro Merino, 2019. "A difference-of-convex functions approach for sparse PDE optimal control problems with nonconvex costs," Computational Optimization and Applications, Springer, vol. 74(1), pages 225-258, September.

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