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On a globally convergent semismooth* Newton method in nonsmooth nonconvex optimization

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  • Helmut Gfrerer

    (Johannes Kepler University Linz
    Austrian Academy of Sciences)

Abstract

In this paper we present GSSN, a globalized SCD semismooth $$^{*}$$ ∗ Newton method for solving nonsmooth nonconvex optimization problems. The global convergence properties of the method are ensured by the proximal gradient method, whereas locally superlinear convergence is established via the SCD semismooth $$^{*}$$ ∗ Newton method under quite weak assumptions. The Newton direction is based on the SC (subspace containing) derivative of the subdifferential mapping and can be computed by the (approximate) solution of an equality-constrained quadratic program. Special attention is given to the efficient numerical implementation of the overall method.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:91:y:2025:i:1:d:10.1007_s10589-025-00658-z
    DOI: 10.1007/s10589-025-00658-z
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    References listed on IDEAS

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    1. 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.
    2. Christian Kanzow & Theresa Lechner, 2021. "Correction to: Globalized inexact proximal Newton-type methods for nonconvex composite functions," Computational Optimization and Applications, Springer, vol. 80(2), pages 679-680, November.
    3. Helmut Gfrerer & Michael Mandlmayr & Jiří V. Outrata & Jan Valdman, 2023. "On the SCD semismooth* Newton method for generalized equations with application to a class of static contact problems with Coulomb friction," Computational Optimization and Applications, Springer, vol. 86(3), pages 1159-1191, December.
    4. 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.
    5. Bastian Pötzl & Anton Schiela & Patrick Jaap, 2022. "Second order semi-smooth Proximal Newton methods in Hilbert spaces," Computational Optimization and Applications, Springer, vol. 82(2), pages 465-498, June.
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