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On the Convergence of the Iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm”

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  • A. Chambolle

    (CMAP, Ecole Polytechnique, CNRS)

  • Ch. Dossal

    (IMB, Univ. Bordeaux 1, CNRS)

Abstract

We discuss here the convergence of the iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm,” which is an algorithm proposed by Beck and Teboulle for minimizing the sum of two convex, lower-semicontinuous, and proper functions (defined in a Euclidean or Hilbert space), such that one is differentiable with Lipschitz gradient, and the proximity operator of the second is easy to compute. It builds a sequence of iterates for which the objective is controlled, up to a (nearly optimal) constant, by the inverse of the square of the iteration number. However, the convergence of the iterates themselves is not known. We show here that with a small modification, we can ensure the same upper bound for the decay of the energy, as well as the convergence of the iterates to a minimizer.

Suggested Citation

  • A. Chambolle & Ch. Dossal, 2015. "On the Convergence of the Iterates of the “Fast Iterative Shrinkage/Thresholding Algorithm”," Journal of Optimization Theory and Applications, Springer, vol. 166(3), pages 968-982, September.
    • Handle: RePEc:spr:joptap:v:166:y:2015:i:3:d:10.1007_s10957-015-0746-4
    DOI: 10.1007/s10957-015-0746-4
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    References listed on IDEAS

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    1. Patrick L. Combettes & Jean-Christophe Pesquet, 2011. "Proximal Splitting Methods in Signal Processing," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 185-212, Springer.
    2. 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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    5. Hongwei Liu & Ting Wang & Zexian Liu, 2022. "Some modified fast iterative shrinkage thresholding algorithms with a new adaptive non-monotone stepsize strategy for nonsmooth and convex minimization problems," Computational Optimization and Applications, Springer, vol. 83(2), pages 651-691, November.
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    10. Lazzaro, D. & Loli Piccolomini, E. & Zama, F., 2019. "A fast splitting method for efficient Split Bregman iterations," Applied Mathematics and Computation, Elsevier, vol. 357(C), pages 139-146.

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