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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
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    References listed on IDEAS

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    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.
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    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.

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