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On the convergence of the iterates of proximal gradient algorithm with extrapolation for convex nonsmooth minimization problems

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  • Bo Wen

    (Hebei University of Technology
    Harbin Institute of Technology)

  • Xiaoping Xue

    (Harbin Institute of Technology
    Harbin Institute of Technology)

Abstract

In this paper, we consider the proximal gradient algorithm with extrapolation for solving a class of convex nonsmooth minimization problems. We show that for a large class of extrapolation parameters including the extrapolation parameters chosen in FISTA (Beck and Teboulle in SIAM J Imaging Sci 2:183–202, 2009), the successive changes of iterates go to 0. Moreover, based on the Łojasiewicz inequality, we establish the global convergence of iterates generated by the proximal gradient algorithm with extrapolation with an additional assumption on the extrapolation coefficients. The assumption is general enough to allow the threshold of the extrapolation coefficients to be 1. In particular, we prove the length of the iterates is finite. Finally, we perform numerical experiments on the least squares problems with $$\ell _1$$ ℓ 1 regularization to illustrate our theoretical results.

Suggested Citation

  • Bo Wen & Xiaoping Xue, 2019. "On the convergence of the iterates of proximal gradient algorithm with extrapolation for convex nonsmooth minimization problems," Journal of Global Optimization, Springer, vol. 75(3), pages 767-787, November.
    • Handle: RePEc:spr:jglopt:v:75:y:2019:i:3:d:10.1007_s10898-019-00789-8
    DOI: 10.1007/s10898-019-00789-8
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    References listed on IDEAS

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    1. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
    2. NESTEROV, Yu., 2007. "Gradient methods for minimizing composite objective function," LIDAM Discussion Papers CORE 2007076, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    3. 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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    Cited by:

    1. Zhili Ge & Zhongming Wu & Xin Zhang & Qin Ni, 2023. "An extrapolated proximal iteratively reweighted method for nonconvex composite optimization problems," Journal of Global Optimization, Springer, vol. 86(4), pages 821-844, August.

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