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Derivative-free separable quadratic modeling and cubic regularization for unconstrained optimization

Author

Listed:
  • A. L. Custódio

    (FCT NOVA
    FCT NOVA)

  • R. Garmanjani

    (FCT NOVA)

  • M. Raydan

    (FCT NOVA)

Abstract

We present a derivative-free separable quadratic modeling and cubic regularization technique for solving smooth unconstrained minimization problems. The derivative-free approach is mainly concerned with building a quadratic model that could be generated by numerical interpolation or using a minimum Frobenius norm approach, when the number of points available does not allow to build a complete quadratic model. This model plays a key role to generate an approximated gradient vector and Hessian matrix of the objective function at every iteration. We add a specialized cubic regularization strategy to minimize the quadratic model at each iteration, that makes use of separability. We discuss convergence results, including worst case complexity, of the proposed schemes to first-order stationary points. Some preliminary numerical results are presented to illustrate the robustness of the specialized separable cubic algorithm.

Suggested Citation

  • A. L. Custódio & R. Garmanjani & M. Raydan, 2024. "Derivative-free separable quadratic modeling and cubic regularization for unconstrained optimization," 4OR, Springer, vol. 22(1), pages 121-144, March.
    • Handle: RePEc:spr:aqjoor:v:22:y:2024:i:1:d:10.1007_s10288-023-00541-9
    DOI: 10.1007/s10288-023-00541-9
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    References listed on IDEAS

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    4. C. P. Brás & J. M. Martínez & M. Raydan, 2020. "Large-scale unconstrained optimization using separable cubic modeling and matrix-free subspace minimization," Computational Optimization and Applications, Springer, vol. 75(1), pages 169-205, January.
    5. 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).
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