IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v189y2021i1d10.1007_s10957-021-01838-7.html
   My bibliography  Save this article

Minimizing Uniformly Convex Functions by Cubic Regularization of Newton Method

Author

Listed:
  • Nikita Doikov

    (ICTEAM (Catholic University of Louvain))

  • Yurii Nesterov

    (CORE (Catholic University of Louvain))

Abstract

In this paper, we study the iteration complexity of cubic regularization of Newton method for solving composite minimization problems with uniformly convex objective. We introduce the notion of second-order condition number of a certain degree and justify the linear rate of convergence in a nondegenerate case for the method with an adaptive estimate of the regularization parameter. The algorithm automatically achieves the best possible global complexity bound among different problem classes of uniformly convex objective functions with Hölder continuous Hessian of the smooth part of the objective. As a byproduct of our developments, we justify an intuitively plausible result that the global iteration complexity of the Newton method is always better than that of the gradient method on the class of strongly convex functions with uniformly bounded second derivative.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:189:y:2021:i:1:d:10.1007_s10957-021-01838-7
    DOI: 10.1007/s10957-021-01838-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-021-01838-7
    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/s10957-021-01838-7?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. 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. 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).
    3. NESTEROV, Yurii, 2007. "Gauss-Newton scheme with worst case guarantees for global performance," LIDAM Reprints CORE 1952, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. Geovani N. Grapiglia & Yurii Nesterov, 2019. "Accelerated regularized Newton methods for minimizing composite convex functions," LIDAM Reprints CORE 3058, 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. Doikov, Nikita & Nesterov, Yurii, 2021. "Optimization Methods for Fully Composite Problems," LIDAM Discussion Papers CORE 2021001, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).

    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. 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.
    2. 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.
    3. 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.
    4. Kenji Ueda & Nobuo Yamashita, 2010. "On a Global Complexity Bound of the Levenberg-Marquardt Method," Journal of Optimization Theory and Applications, Springer, vol. 147(3), pages 443-453, December.
    5. 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.
    6. Boris Polyak & Andrey Tremba, 2020. "Sparse solutions of optimal control via Newton method for under-determined systems," Journal of Global Optimization, Springer, vol. 76(3), pages 613-623, March.
    7. 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.
    8. 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).
    9. 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.
    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. 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.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. Yakui Huang & Hongwei Liu, 2016. "Smoothing projected Barzilai–Borwein method for constrained non-Lipschitz optimization," Computational Optimization and Applications, Springer, vol. 65(3), pages 671-698, December.
    18. Nadav Hallak & Marc Teboulle, 2020. "Finding Second-Order Stationary Points in Constrained Minimization: A Feasible Direction Approach," Journal of Optimization Theory and Applications, Springer, vol. 186(2), pages 480-503, August.
    19. Francisco Facchinei & Vyacheslav Kungurtsev & Lorenzo Lampariello & Gesualdo Scutari, 2021. "Ghost Penalties in Nonconvex Constrained Optimization: Diminishing Stepsizes and Iteration Complexity," Mathematics of Operations Research, INFORMS, vol. 46(2), pages 595-627, May.
    20. Rujun Jiang & Man-Chung Yue & Zhishuo Zhou, 2021. "An accelerated first-order method with complexity analysis for solving cubic regularization subproblems," Computational Optimization and Applications, Springer, vol. 79(2), pages 471-506, June.

    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:joptap:v:189:y:2021:i:1:d:10.1007_s10957-021-01838-7. 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.