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Non-smooth Non-convex Bregman Minimization: Unification and New Algorithms

Author

Listed:
  • Peter Ochs

    (Saarland University)

  • Jalal Fadili

    (Normandie Université ENSICAEN, CNRS, GREYC)

  • Thomas Brox

    (University of Freiburg)

Abstract

We propose a unifying algorithm for non-smooth non-convex optimization. The algorithm approximates the objective function by a convex model function and finds an approximate (Bregman) proximal point of the convex model. This approximate minimizer of the model function yields a descent direction, along which the next iterate is found. Complemented with an Armijo-like line search strategy, we obtain a flexible algorithm for which we prove (subsequential) convergence to a stationary point under weak assumptions on the growth of the model function error. Special instances of the algorithm with a Euclidean distance function are, for example, gradient descent, forward–backward splitting, ProxDescent, without the common requirement of a “Lipschitz continuous gradient”. In addition, we consider a broad class of Bregman distance functions (generated by Legendre functions), replacing the Euclidean distance. The algorithm has a wide range of applications including many linear and nonlinear inverse problems in signal/image processing and machine learning.

Suggested Citation

  • Peter Ochs & Jalal Fadili & Thomas Brox, 2019. "Non-smooth Non-convex Bregman Minimization: Unification and New Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 181(1), pages 244-278, April.
    • Handle: RePEc:spr:joptap:v:181:y:2019:i:1:d:10.1007_s10957-018-01452-0
    DOI: 10.1007/s10957-018-01452-0
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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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    Cited by:

    1. Bonettini, S. & Prato, M. & Rebegoldi, S., 2021. "New convergence results for the inexact variable metric forward–backward method," Applied Mathematics and Computation, Elsevier, vol. 392(C).
    2. Mahesh Chandra Mukkamala & Jalal Fadili & Peter Ochs, 2022. "Global convergence of model function based Bregman proximal minimization algorithms," Journal of Global Optimization, Springer, vol. 83(4), pages 753-781, August.
    3. Zhongming Wu & Chongshou Li & Min Li & Andrew Lim, 2021. "Inertial proximal gradient methods with Bregman regularization for a class of nonconvex optimization problems," Journal of Global Optimization, Springer, vol. 79(3), pages 617-644, March.

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