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Large-scale unconstrained optimization using separable cubic modeling and matrix-free subspace minimization

Author

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  • C. P. Brás

    (UNL
    UNL)

  • J. M. Martínez

    (University of Campinas)

  • M. Raydan

    (UNL)

Abstract

We present a new algorithm for solving large-scale unconstrained optimization problems that uses cubic models, matrix-free subspace minimization, and secant-type parameters for defining the cubic terms. We also propose and analyze a specialized trust-region strategy to minimize the cubic model on a properly chosen low-dimensional subspace, which is built at each iteration using the Lanczos process. For the convergence analysis we present, as a general framework, a model trust-region subspace algorithm with variable metric and we establish asymptotic as well as complexity convergence results. Preliminary numerical results, on some test functions and also on the well-known disk packing problem, are presented to illustrate the performance of the proposed scheme when solving large-scale problems.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:75:y:2020:i:1:d:10.1007_s10589-019-00138-1
    DOI: 10.1007/s10589-019-00138-1
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    References listed on IDEAS

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    1. El Houcine Bergou & Youssef Diouane & Serge Gratton, 2018. "A Line-Search Algorithm Inspired by the Adaptive Cubic Regularization Framework and Complexity Analysis," Journal of Optimization Theory and Applications, Springer, vol. 178(3), pages 885-913, September.
    2. E. Bergou & Y. Diouane & S. Gratton, 2017. "On the use of the energy norm in trust-region and adaptive cubic regularization subproblems," Computational Optimization and Applications, Springer, vol. 68(3), pages 533-554, December.
    3. Birgin, E. G. & Martinez, J. M. & Ronconi, D. P., 2005. "Optimizing the packing of cylinders into a rectangular container: A nonlinear approach," European Journal of Operational Research, Elsevier, vol. 160(1), pages 19-33, January.
    4. 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.
    5. 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.
    6. GRAPIGLIA, Geovani Nunes & NESTEROV, Yurii, 2016. "Globally Convergent Second-order Schemes for Minimizing Twice-differentiable Functions," LIDAM Discussion Papers CORE 2016028, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    7. Giovanni Fasano & Massimo Roma, 2013. "Preconditioning Newton–Krylov methods in nonconvex large scale optimization," Computational Optimization and Applications, Springer, vol. 56(2), pages 253-290, October.
    8. 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.
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    Cited by:

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