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A Laplacian approach to $$\ell _1$$ ℓ 1 -norm minimization

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  • Vincenzo Bonifaci

    (Università degli Studi Roma Tre)

Abstract

We propose a novel differentiable reformulation of the linearly-constrained $$\ell _1$$ ℓ 1 minimization problem, also known as the basis pursuit problem. The reformulation is inspired by the Laplacian paradigm of network theory and leads to a new family of gradient-based methods for the solution of $$\ell _1$$ ℓ 1 minimization problems. We analyze the iteration complexity of a natural solution approach to the reformulation, based on a multiplicative weights update scheme, as well as the iteration complexity of an accelerated gradient scheme. The results can be seen as bounds on the complexity of iteratively reweighted least squares (IRLS) type methods of basis pursuit.

Suggested Citation

  • Vincenzo Bonifaci, 2021. "A Laplacian approach to $$\ell _1$$ ℓ 1 -norm minimization," Computational Optimization and Applications, Springer, vol. 79(2), pages 441-469, June.
    • Handle: RePEc:spr:coopap:v:79:y:2021:i:2:d:10.1007_s10589-021-00270-x
    DOI: 10.1007/s10589-021-00270-x
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    References listed on IDEAS

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    1. Haihao Lu & Robert M. Freund & Yurii Nesterov, 2018. "Relatively smooth convex optimization by first-order methods, and applications," LIDAM Reprints CORE 2965, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Yurii Nesterov, 2018. "Smooth Convex Optimization," Springer Optimization and Its Applications, in: Lectures on Convex Optimization, edition 2, chapter 0, pages 59-137, Springer.
    3. Heinz H. Bauschke & Jérôme Bolte & Marc Teboulle, 2017. "A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 330-348, May.
    4. Saha, Tanay & Srivastava, Shwetabh & Khare, Swanand & Stanimirović, Predrag S. & Petković, Marko D., 2019. "An improved algorithm for basis pursuit problem and its applications," Applied Mathematics and Computation, Elsevier, vol. 355(C), pages 385-398.
    5. NESTEROV, Yu., 2005. "Smooth minimization of non-smooth functions," LIDAM Reprints CORE 1819, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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