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A regularized Newton method without line search for unconstrained optimization

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  • Kenji Ueda
  • Nobuo Yamashita

Abstract

In this paper, we propose a regularized Newton method without line search. The proposed method controls a regularization parameter instead of a step size in order to guarantee the global convergence. We show that the proposed algorithm has the following convergence properties. (a) The proposed algorithm has global convergence under appropriate conditions. (b) It has superlinear rate of convergence under the local error bound condition. (c) An upper bound of the number of iterations required to obtain an approximate solution $$x$$ x satisfying $$\Vert \nabla f(x) \Vert \le \varepsilon $$ ‖ ∇ f ( x ) ‖ ≤ ε is $$O(\varepsilon ^{-2})$$ O ( ε - 2 ) , where $$f$$ f is the objective function and $$\varepsilon $$ ε is a given positive constant. Copyright Springer Science+Business Media New York 2014

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  • Kenji Ueda & Nobuo Yamashita, 2014. "A regularized Newton method without line search for unconstrained optimization," Computational Optimization and Applications, Springer, vol. 59(1), pages 321-351, October.
    • Handle: RePEc:spr:coopap:v:59:y:2014:i:1:p:321-351
    DOI: 10.1007/s10589-014-9656-x
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    References listed on IDEAS

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    1. NESTEROV, Yurii & POLYAK, B.T., 2006. "Cubic regularization of Newton method and its global performance," LIDAM Reprints CORE 1927, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Ying-Jie Li & Dong-Hui Li, 2009. "Truncated regularized Newton method for convex minimizations," Computational Optimization and Applications, Springer, vol. 43(1), pages 119-131, May.
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    Cited by:

    1. Paul Armand & Ngoc Nguyen Tran, 2021. "Local Convergence Analysis of a Primal–Dual Method for Bound-Constrained Optimization Without SOSC," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 96-116, April.
    2. Tuyen Trung Truong & Tat Dat To & Hang-Tuan Nguyen & Thu Hang Nguyen & Hoang Phuong Nguyen & Maged Helmy, 2023. "A Fast and Simple Modification of Newton’s Method Avoiding Saddle Points," Journal of Optimization Theory and Applications, Springer, vol. 199(2), pages 805-830, November.
    3. Jianyu Xiao & Haibin Zhang & Huan Gao, 2025. "A Chebyshev–Halley Method with Gradient Regularization and an Improved Convergence Rate," Mathematics, MDPI, vol. 13(8), pages 1-17, April.
    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. Paul Armand & Isaï Lankoandé, 2017. "An inexact proximal regularization method for unconstrained optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(1), pages 43-59, February.

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