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Minimization over the $$\ell _1$$ ℓ 1 -ball using an active-set non-monotone projected gradient

Author

Listed:
  • Andrea Cristofari

    (Università degli Studi di Roma “Tor Vergata”)

  • Marianna Santis

    (Sapienza Università di Roma)

  • Stefano Lucidi

    (Sapienza Università di Roma)

  • Francesco Rinaldi

    (Università di Padova)

Abstract

The $$\ell _1$$ ℓ 1 -ball is a nicely structured feasible set that is widely used in many fields (e.g., machine learning, statistics and signal analysis) to enforce some sparsity in the model solutions. In this paper, we devise an active-set strategy for efficiently dealing with minimization problems over the $$\ell _1$$ ℓ 1 -ball and embed it into a tailored algorithmic scheme that makes use of a non-monotone first-order approach to explore the given subspace at each iteration. We prove global convergence to stationary points. Finally, we report numerical experiments, on two different classes of instances, showing the effectiveness of the algorithm.

Suggested Citation

  • Andrea Cristofari & Marianna Santis & Stefano Lucidi & Francesco Rinaldi, 2022. "Minimization over the $$\ell _1$$ ℓ 1 -ball using an active-set non-monotone projected gradient," Computational Optimization and Applications, Springer, vol. 83(2), pages 693-721, November.
    • Handle: RePEc:spr:coopap:v:83:y:2022:i:2:d:10.1007_s10589-022-00407-6
    DOI: 10.1007/s10589-022-00407-6
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    References listed on IDEAS

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    1. Brás, Carmo P. & Fischer, Andreas & Júdice, Joaquim J. & Schönefeld, Klaus & Seifert, Sarah, 2017. "A block active set algorithm with spectral choice line search for the symmetric eigenvalue complementarity problem," Applied Mathematics and Computation, Elsevier, vol. 294(C), pages 36-48.
    2. Andrea Cristofari & Marianna Santis & Stefano Lucidi & Francesco Rinaldi, 2020. "An active-set algorithmic framework for non-convex optimization problems over the simplex," Computational Optimization and Applications, Springer, vol. 77(1), pages 57-89, September.
    3. Andrea Cristofari & Marianna Santis & Stefano Lucidi & Francesco Rinaldi, 2017. "A Two-Stage Active-Set Algorithm for Bound-Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 172(2), pages 369-401, February.
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    Cited by:

    1. Cristofari, Andrea, 2023. "A decomposition method for lasso problems with zero-sum constraint," European Journal of Operational Research, Elsevier, vol. 306(1), pages 358-369.

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