Skip to main content
Springer Nature Link
Log in

Some Properties of Smooth Convex Functions and Newton’s Method

  • MATHEMATICS
  • Published:
Doklady Mathematics Aims and scope Submit manuscript

Abstract

New properties of convex infinitely differentiable functions related to extremal problems are established. It is shown that, in a neighborhood of the solution, even if the Hessian matrix is singular at the solution point of the function to be minimized, the gradient of the objective function belongs to the image of its second derivative. Due to this new property of convex functions, Newtonian methods for solving unconstrained optimization problems can be applied without assuming the nonsingularity of the Hessian matrix at the solution of the problem and their rate of convergence in argument can be estimated under fairly general assumptions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. B. T. Polyak, Tr. Inst. Sist. Anal. Ross. Akad. Nauk 28, 44–62 (2006).

    Google Scholar 

  2. B. Bamadio and K. A. Lebedev, Mezhdunar. Nauch.-Issled. Zh. 37 (6–2), 11–14 (2015).

  3. V. I. Zabotin and Yu. A. Chernyaev, Comput. Math. Math. Phys. 58 (3), 322–327 (2018). https://doi.org/10.1134/S0965542518030144

    Article  MathSciNet  Google Scholar 

  4. D. Budzko, A. Cordero, and J. R. Torregrosa, SeMA J. 66 (1), 43–53 (2014). https://doi.org/10.1007/s40324-014-0020-y

    Article  MathSciNet  Google Scholar 

  5. Y. Nesterov, Math. Program. 112 (1), 159–181 (2008). https://doi.org/10.1007/s10107-006-0089-x

    Article  MathSciNet  Google Scholar 

  6. B. Polyak and A. Tremba, Optim. Methods Software 35 (6), 1272–1303 (2020). https://doi.org/10.1080/10556788.2019.1669154

    Article  MathSciNet  Google Scholar 

  7. T. H. Colding and W. P. Minicozzi, “Lojasiewicz inequalities and applications,” arXiv:1402.5087. 2014.

  8. S. Lojasiewicz, C. R. Acad. Sci. 246 (5), 683–686 (1958).

    Google Scholar 

Download references

Funding

This work was supported by the Russian Science Foundation, project no. 21-71-30005.

Author information

Authors and Affiliations

  1. Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, 119991, Moscow, Russia

    D. V. Denisov &  Yu. G. Evtushenko

  2. Dorodnicyn Computing Centre, Federal Research Center “Computer Science and Control,” Russian Academy of Sciences, 119333, Moscow, Russia

    Yu. G. Evtushenko & A. A. Tret’yakov

  3. Moscow Institute of Physics and Technology (National Research University), 141700, Dolgoprudnyi, Moscow oblast, Russia

    Yu. G. Evtushenko

  4. Moscow Aviation Institute (National Research University), 125080, Moscow, Russia

    Yu. G. Evtushenko

  5. System Research Institute, Polish Academy of Sciences, 01-447, Warsaw, Poland

    A. A. Tret’yakov

  6. Faculty of Sciences, Siedlce University, 08-110, Siedlce, Poland

    A. A. Tret’yakov

Authors
  1. D. V. Denisov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu. G. Evtushenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. A. Tret’yakov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to D. V. Denisov, Yu. G. Evtushenko or A. A. Tret’yakov.

Additional information

Translated by I. Ruzanova

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Denisov, D.V., Evtushenko, Y.G. & Tret’yakov, A.A. Some Properties of Smooth Convex Functions and Newton’s Method. Dokl. Math. 103, 76–80 (2021). https://doi.org/10.1134/S1064562421020034

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1064562421020034

Keywords:

Search

Navigation