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Separable cubic modeling and a trust-region strategy for unconstrained minimization with impact in global optimization

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  • J. Martínez
  • M. Raydan

Abstract

A separable cubic model, for smooth unconstrained minimization, is proposed and evaluated. The cubic model uses some novel secant-type choices for the parameters in the cubic terms. A suitable hard-case-free trust-region strategy that takes advantage of the separable cubic modeling is also presented. For the convergence analysis of our specialized trust region strategy we present as a general framework a model $$q$$ q -order trust region algorithm with variable metric and we prove its convergence to $$q$$ q -stationary points. Some preliminary numerical examples are also presented to illustrate the tendency of the specialized trust region algorithm, when combined with our cubic modeling, to escape from local minimizers. Copyright Springer Science+Business Media New York 2015

Suggested Citation

  • J. Martínez & M. Raydan, 2015. "Separable cubic modeling and a trust-region strategy for unconstrained minimization with impact in global optimization," Journal of Global Optimization, Springer, vol. 63(2), pages 319-342, October.
    • Handle: RePEc:spr:jglopt:v:63:y:2015:i:2:p:319-342
    DOI: 10.1007/s10898-015-0278-3
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    References listed on IDEAS

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    1. Birgin, Ernesto G. & Martínez, Jose Mario & Raydan, Marcos, 2014. "Spectral Projected Gradient Methods: Review and Perspectives," Journal of Statistical Software, Foundation for Open Access Statistics, vol. 60(i03).
    2. Sha Lu & Zengxin Wei & Lue Li, 2012. "A trust region algorithm with adaptive cubic regularization methods for nonsmooth convex minimization," Computational Optimization and Applications, Springer, vol. 51(2), pages 551-573, March.
    3. Hande Benson & David Shanno, 2014. "Interior-point methods for nonconvex nonlinear programming: cubic regularization," Computational Optimization and Applications, Springer, vol. 58(2), pages 323-346, June.
    4. N. Gould & M. Porcelli & P. Toint, 2012. "Updating the regularization parameter in the adaptive cubic regularization algorithm," Computational Optimization and Applications, Springer, vol. 53(1), pages 1-22, September.
    5. Tommaso Bianconcini & Giampaolo Liuzzi & Benedetta Morini & Marco Sciandrone, 2015. "On the use of iterative methods in cubic regularization for unconstrained optimization," Computational Optimization and Applications, Springer, vol. 60(1), pages 35-57, January.
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    1. J. M. Martínez & M. Raydan, 2017. "Cubic-regularization counterpart of a variable-norm trust-region method for unconstrained minimization," Journal of Global Optimization, Springer, vol. 68(2), pages 367-385, June.
    2. V. S. Amaral & R. Andreani & E. G. Birgin & D. S. Marcondes & J. M. Martínez, 2022. "On complexity and convergence of high-order coordinate descent algorithms for smooth nonconvex box-constrained minimization," Journal of Global Optimization, Springer, vol. 84(3), pages 527-561, November.
    3. 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.

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