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A Telescopic Bregmanian Proximal Gradient Method Without the Global Lipschitz Continuity Assumption

Author

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  • Daniel Reem

    (The Technion - Israel Institute of Technology)

  • Simeon Reich

    (The Technion - Israel Institute of Technology)

  • Alvaro Pierro

    (CNPq)

Abstract

The problem of minimization of the sum of two convex functions has various theoretical and real-world applications. One of the popular methods for solving this problem is the proximal gradient method (proximal forward–backward algorithm). A very common assumption in the use of this method is that the gradient of the smooth term is globally Lipschitz continuous. However, this assumption is not always satisfied in practice, thus casting a limitation on the method. In this paper, we discuss, in a wide class of finite- and infinite-dimensional spaces, a new variant of the proximal gradient method, which does not impose the above-mentioned global Lipschitz continuity assumption. A key contribution of the method is the dependence of the iterative steps on a certain telescopic decomposition of the constraint set into subsets. Moreover, we use a Bregman divergence in the proximal forward–backward operation. Under certain practical conditions, a non-asymptotic rate of convergence (that is, in the function values) is established, as well as the weak convergence of the whole sequence to a minimizer. We also obtain a few auxiliary results of independent interest.

Suggested Citation

  • Daniel Reem & Simeon Reich & Alvaro Pierro, 2019. "A Telescopic Bregmanian Proximal Gradient Method Without the Global Lipschitz Continuity Assumption," Journal of Optimization Theory and Applications, Springer, vol. 182(3), pages 851-884, September.
    • Handle: RePEc:spr:joptap:v:182:y:2019:i:3:d:10.1007_s10957-019-01509-8
    DOI: 10.1007/s10957-019-01509-8
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    References listed on IDEAS

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    1. 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.
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    1. Emanuel Laude & Peter Ochs & Daniel Cremers, 2020. "Bregman Proximal Mappings and Bregman–Moreau Envelopes Under Relative Prox-Regularity," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 724-761, March.

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