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Quadratic regularization methods with finite-difference gradient approximations

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  • Geovani Nunes Grapiglia

    (Université catholique de Louvain, ICTEAM/INMA)

Abstract

This paper presents two quadratic regularization methods with finite-difference gradient approximations for smooth unconstrained optimization problems. One method is based on forward finite-difference gradients, while the other is based on central finite-difference gradients. In both methods, the accuracy of the gradient approximations and the regularization parameter in the quadratic models are jointly adjusted using a nonmonotone acceptance condition for the trial points. When the objective function is bounded from below and has Lipschitz continuous gradient, it is shown that the method based on forward finite-difference gradients needs at most $${\mathcal{O}}\left( n\epsilon ^{-2}\right) $$ O n ϵ - 2 function evaluations to generate a $$\epsilon $$ ϵ -approximate stationary point, where n is the problem dimension. Under the additional assumption that the Hessian of the objective is Lipschitz continuous, an evaluation complexity bound of the same order is proved for the method based on central finite-difference gradients. Numerical results are also presented. They confirm the theoretical findings and illustrate the relative efficiency of the proposed methods.

Suggested Citation

  • Geovani Nunes Grapiglia, 2023. "Quadratic regularization methods with finite-difference gradient approximations," Computational Optimization and Applications, Springer, vol. 85(3), pages 683-703, July.
    • Handle: RePEc:spr:coopap:v:85:y:2023:i:3:d:10.1007_s10589-022-00373-z
    DOI: 10.1007/s10589-022-00373-z
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    References listed on IDEAS

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    1. Geovani N. Grapiglia & Ekkehard W. Sachs, 2017. "On the worst-case evaluation complexity of non-monotone line search algorithms," Computational Optimization and Applications, Springer, vol. 68(3), pages 555-577, December.
    2. Yurii NESTEROV & Vladimir SPOKOINY, 2017. "Random gradient-free minimization of convex functions," LIDAM Reprints CORE 2851, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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