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Globalized inexact proximal Newton-type methods for nonconvex composite functions

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  • Christian Kanzow

    (University of Würzburg, Institute of Mathematics)

  • Theresa Lechner

    (University of Würzburg, Institute of Mathematics)

Abstract

Optimization problems with composite functions consist of an objective function which is the sum of a smooth and a (convex) nonsmooth term. This particular structure is exploited by the class of proximal gradient methods and some of their generalizations like proximal Newton and quasi-Newton methods. The current literature on these classes of methods almost exclusively considers the case where also the smooth term is convex. Here we present a globalized proximal Newton-type method which allows the smooth term to be nonconvex. The method is shown to have nice global and local convergence properties, and some numerical results indicate that this method is very promising also from a practical point of view.

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  • Christian Kanzow & Theresa Lechner, 2021. "Globalized inexact proximal Newton-type methods for nonconvex composite functions," Computational Optimization and Applications, Springer, vol. 78(2), pages 377-410, March.
    • Handle: RePEc:spr:coopap:v:78:y:2021:i:2:d:10.1007_s10589-020-00243-6
    DOI: 10.1007/s10589-020-00243-6
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    References listed on IDEAS

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    1. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Hiva Ghanbari & Katya Scheinberg, 2018. "Proximal quasi-Newton methods for regularized convex optimization with linear and accelerated sublinear convergence rates," Computational Optimization and Applications, Springer, vol. 69(3), pages 597-627, April.
    3. Ching-pei Lee & Stephen J. Wright, 2019. "Inexact Successive quadratic approximation for regularized optimization," Computational Optimization and Applications, Springer, vol. 72(3), pages 641-674, April.
    4. Kimon Fountoulakis & Rachael Tappenden, 2018. "A flexible coordinate descent method," Computational Optimization and Applications, Springer, vol. 70(2), pages 351-394, June.
    5. Ming Yuan & Yi Lin, 2006. "Model selection and estimation in regression with grouped variables," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 68(1), pages 49-67, February.
    6. Lorenzo Stella & Andreas Themelis & Panagiotis Patrinos, 2017. "Forward–backward quasi-Newton methods for nonsmooth optimization problems," Computational Optimization and Applications, Springer, vol. 67(3), pages 443-487, July.
    7. P. Tseng & S. Yun, 2009. "Block-Coordinate Gradient Descent Method for Linearly Constrained Nonsmooth Separable Optimization," Journal of Optimization Theory and Applications, Springer, vol. 140(3), pages 513-535, March.
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    Cited by:

    1. Christian Kanzow & Theresa Lechner, 2022. "COAP 2021 Best Paper Prize," Computational Optimization and Applications, Springer, vol. 83(3), pages 723-726, December.
    2. Behzad Azmi & Marco Bernreuther, 2025. "On the forward–backward method with nonmonotone linesearch for infinite-dimensional nonsmooth nonconvex problems," Computational Optimization and Applications, Springer, vol. 91(3), pages 1263-1308, July.
    3. Ruyu Liu & Shaohua Pan & Yuqia Wu & Xiaoqi Yang, 2024. "An inexact regularized proximal Newton method for nonconvex and nonsmooth optimization," Computational Optimization and Applications, Springer, vol. 88(2), pages 603-641, June.
    4. Simeon vom Dahl & Christian Kanzow, 2024. "An inexact regularized proximal Newton method without line search," Computational Optimization and Applications, Springer, vol. 89(3), pages 585-624, December.
    5. Helmut Gfrerer, 2025. "On a globally convergent semismooth* Newton method in nonsmooth nonconvex optimization," Computational Optimization and Applications, Springer, vol. 91(1), pages 67-124, May.

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