IDEAS home Printed from https://ideas.repec.org/a/spr/eurjco/v5y2017i4d10.1007_s13675-017-0081-7.html
   My bibliography  Save this article

On the construction of quadratic models for derivative-free trust-region algorithms

Author

Listed:
  • Adriano Verdério

    (Federal University of Technology - Paraná)

  • Elizabeth W. Karas

    (Federal University of Paraná)

  • Lucas G. Pedroso

    (Federal University of Paraná)

  • Katya Scheinberg

    (Lehigh University)

Abstract

We consider derivative-free trust-region algorithms based on sampling approaches for convex constrained problems and discuss two conditions on the quadratic models for ensuring their global convergence. The first condition requires the poisedness of the sample sets, as usual in this context, while the other one is related to the error between the model and the objective function at the sample points. Although the second condition trivially holds if the model is constructed by polynomial interpolation, since in this case the model coincides with the objective function at the sample set, we show that it also holds for models constructed by support vector regression. These two conditions imply that the error between the gradient of the trust-region model and the objective function is of the order of $$\delta _k$$ δ k , where $$\delta _k$$ δ k controls the diameter of the sample set. This allows proving the global convergence of a trust-region algorithm that uses two radii, $$\delta _k$$ δ k and the trust-region radius. Preliminary numerical experiments are presented for minimizing functions with and without noise.

Suggested Citation

  • Adriano Verdério & Elizabeth W. Karas & Lucas G. Pedroso & Katya Scheinberg, 2017. "On the construction of quadratic models for derivative-free trust-region algorithms," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 5(4), pages 501-527, December.
    • Handle: RePEc:spr:eurjco:v:5:y:2017:i:4:d:10.1007_s13675-017-0081-7
    DOI: 10.1007/s13675-017-0081-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s13675-017-0081-7
    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/s13675-017-0081-7?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. M. Powell, 2012. "On the convergence of trust region algorithms for unconstrained minimization without derivatives," Computational Optimization and Applications, Springer, vol. 53(2), pages 527-555, October.
    2. E. Gumma & M. Hashim & M. Ali, 2014. "A derivative-free algorithm for linearly constrained optimization problems," Computational Optimization and Applications, Springer, vol. 57(3), pages 599-621, April.
    3. Ernesto Birgin & Jan Gentil, 2012. "Evaluating bound-constrained minimization software," Computational Optimization and Applications, Springer, vol. 53(2), pages 347-373, October.
    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. Charles Audet & Sébastien Le Digabel & Renaud Saltet, 2022. "Quantifying uncertainty with ensembles of surrogates for blackbox optimization," Computational Optimization and Applications, Springer, vol. 83(1), pages 29-66, September.
    2. Butyn, Emerson & Karas, Elizabeth W. & de Oliveira, Welington, 2022. "A derivative-free trust-region algorithm with copula-based models for probability maximization problems," European Journal of Operational Research, Elsevier, vol. 298(1), pages 59-75.

    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. Ubaldo M. García Palomares, 2023. "Convergence of derivative-free nonmonotone Direct Search Methods for unconstrained and box-constrained mixed-integer optimization," Computational Optimization and Applications, Springer, vol. 85(3), pages 821-856, July.
    2. Elizabeth W. Karas & Sandra A. Santos & Benar F. Svaiter, 2016. "Algebraic rules for computing the regularization parameter of the Levenberg–Marquardt method," Computational Optimization and Applications, Springer, vol. 65(3), pages 723-751, December.
    3. 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.
    4. Xiuhua Wang & Jisheng Kou, 2015. "Convergence analysis on a class of improved Chebyshev methods for nonlinear equations in Banach spaces," Computational Optimization and Applications, Springer, vol. 60(3), pages 697-717, April.
    5. Charles Audet & Andrew R. Conn & Sébastien Le Digabel & Mathilde Peyrega, 2018. "A progressive barrier derivative-free trust-region algorithm for constrained optimization," Computational Optimization and Applications, Springer, vol. 71(2), pages 307-329, November.
    6. Elizabeth Karas & Sandra Santos & Benar Svaiter, 2015. "Algebraic rules for quadratic regularization of Newton’s method," Computational Optimization and Applications, Springer, vol. 60(2), pages 343-376, March.
    7. Andrea Cristofari & Marianna Santis & Stefano Lucidi & Francesco Rinaldi, 2017. "A Two-Stage Active-Set Algorithm for Bound-Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 172(2), pages 369-401, February.

    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:eurjco:v:5:y:2017:i:4:d:10.1007_s13675-017-0081-7. 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.