IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v73y2019i1d10.1007_s10589-019-00064-2.html
   My bibliography  Save this article

Convergence and evaluation-complexity analysis of a regularized tensor-Newton method for solving nonlinear least-squares problems

Author

Listed:
  • Nicholas I. M. Gould

    (STFC Rutherford Appleton Laboratory)

  • Tyrone Rees

    (STFC Rutherford Appleton Laboratory)

  • Jennifer A. Scott

    (STFC Rutherford Appleton Laboratory
    University of Reading)

Abstract

Given a twice-continuously differentiable vector-valued function r(x), a local minimizer of $$\Vert r(x)\Vert _2$$ ‖ r ( x ) ‖ 2 is sought. We propose and analyse tensor-Newton methods, in which r(x) is replaced locally by its second-order Taylor approximation. Convergence is controlled by regularization of various orders. We establish global convergence to a first-order critical point of $$\Vert r(x)\Vert _2$$ ‖ r ( x ) ‖ 2 , and provide function evaluation bounds that agree with the best-known bounds for methods using second derivatives. Numerical experiments comparing tensor-Newton methods with regularized Gauss–Newton and Newton methods demonstrate the practical performance of the newly proposed method.

Suggested Citation

  • Nicholas I. M. Gould & Tyrone Rees & Jennifer A. Scott, 2019. "Convergence and evaluation-complexity analysis of a regularized tensor-Newton method for solving nonlinear least-squares problems," Computational Optimization and Applications, Springer, vol. 73(1), pages 1-35, May.
  • Handle: RePEc:spr:coopap:v:73:y:2019:i:1:d:10.1007_s10589-019-00064-2
    DOI: 10.1007/s10589-019-00064-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-019-00064-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10589-019-00064-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Geovani N. GRAPIGLIA & Yurii NESTEROV, 2017. "Regularized Newton methods for minimizing functions with Hölder continuous Hessians," LIDAM Reprints CORE 2846, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Hongchao Zhang & Andrew Conn, 2012. "On the local convergence of a derivative-free algorithm for least-squares minimization," Computational Optimization and Applications, Springer, vol. 51(2), pages 481-507, March.
    3. 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).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Xihua Zhu & Jiangze Han & Bo Jiang, 2022. "An adaptive high order method for finding third-order critical points of nonconvex optimization," Journal of Global Optimization, Springer, vol. 84(2), pages 369-392, October.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. J. M. Martínez & L. T. Santos, 2022. "On large-scale unconstrained optimization and arbitrary regularization," Computational Optimization and Applications, Springer, vol. 81(1), pages 1-30, January.
    2. Anton Rodomanov & Yurii Nesterov, 2020. "Smoothness Parameter of Power of Euclidean Norm," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 303-326, May.
    3. Nikita Doikov & Yurii Nesterov, 2021. "Minimizing Uniformly Convex Functions by Cubic Regularization of Newton Method," Journal of Optimization Theory and Applications, Springer, vol. 189(1), pages 317-339, April.
    4. Silvia Berra & Alessandro Torraca & Federico Benvenuto & Sara Sommariva, 2024. "Combined Newton-Gradient Method for Constrained Root-Finding in Chemical Reaction Networks," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 404-427, January.
    5. Ariizumi, Shumpei & Yamakawa, Yuya & Yamashita, Nobuo, 2024. "Convergence properties of Levenberg–Marquardt methods with generalized regularization terms," Applied Mathematics and Computation, Elsevier, vol. 463(C).
    6. Seonho Park & Seung Hyun Jung & Panos M. Pardalos, 2020. "Combining Stochastic Adaptive Cubic Regularization with Negative Curvature for Nonconvex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 953-971, March.
    7. Weiwei Kong & Jefferson G. Melo & Renato D. C. Monteiro, 2020. "An efficient adaptive accelerated inexact proximal point method for solving linearly constrained nonconvex composite problems," Computational Optimization and Applications, Springer, vol. 76(2), pages 305-346, June.
    8. Geovani Nunes Grapiglia & Jinyun Yuan & Ya-xiang Yuan, 2016. "Nonlinear Stepsize Control Algorithms: Complexity Bounds for First- and Second-Order Optimality," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 980-997, December.
    9. Wang, Peng & Zhu, Detong, 2016. "An inexact derivative-free Levenberg–Marquardt method for linear inequality constrained nonlinear systems under local error bound conditions," Applied Mathematics and Computation, Elsevier, vol. 282(C), pages 32-52.
    10. Kenji Ueda & Nobuo Yamashita, 2012. "Global Complexity Bound Analysis of the Levenberg–Marquardt Method for Nonsmooth Equations and Its Application to the Nonlinear Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 152(2), pages 450-467, February.
    11. E. G. Birgin & J. M. Martínez, 2019. "A Newton-like method with mixed factorizations and cubic regularization for unconstrained minimization," Computational Optimization and Applications, Springer, vol. 73(3), pages 707-753, July.
    12. 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.
    13. Liaoyuan Zeng & Ting Kei Pong, 2022. "$$\rho$$ ρ -regularization subproblems: strong duality and an eigensolver-based algorithm," Computational Optimization and Applications, Springer, vol. 81(2), pages 337-368, March.
    14. Yuning Jiang & Dimitris Kouzoupis & Haoyu Yin & Moritz Diehl & Boris Houska, 2021. "Decentralized Optimization Over Tree Graphs," Journal of Optimization Theory and Applications, Springer, vol. 189(2), pages 384-407, May.
    15. Nesterov, Yurii, 2022. "Quartic Regularity," LIDAM Discussion Papers CORE 2022001, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    16. Fedor Stonyakin & Ilya Kuruzov & Boris Polyak, 2023. "Stopping Rules for Gradient Methods for Non-convex Problems with Additive Noise in Gradient," Journal of Optimization Theory and Applications, Springer, vol. 198(2), pages 531-551, August.
    17. Yuquan Chen & Yunkang Sun & Bing Wang, 2023. "Improving the Performance of Optimization Algorithms Using the Adaptive Fixed-Time Scheme and Reset Scheme," Mathematics, MDPI, vol. 11(22), pages 1-16, November.
    18. Yurii Nesterov, 2021. "Superfast Second-Order Methods for Unconstrained Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 1-30, October.
    19. V. S. Amaral & R. Andreani & E. G. Birgin & D. S. Marcondes & J. M. Martínez, 2022. "On complexity and convergence of high-order coordinate descent algorithms for smooth nonconvex box-constrained minimization," Journal of Global Optimization, Springer, vol. 84(3), pages 527-561, November.
    20. Jaroslav Fowkes & Nicholas Gould & Chris Farmer, 2013. "A branch and bound algorithm for the global optimization of Hessian Lipschitz continuous functions," Journal of Global Optimization, Springer, vol. 56(4), pages 1791-1815, August.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:coopap:v:73:y:2019:i:1:d:10.1007_s10589-019-00064-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.