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On complexity and convergence of high-order coordinate descent algorithms for smooth nonconvex box-constrained minimization

Author

Listed:
  • V. S. Amaral

    (University of Campinas)

  • R. Andreani

    (University of Campinas)

  • E. G. Birgin

    (University of São Paulo)

  • D. S. Marcondes

    (University of São Paulo)

  • J. M. Martínez

    (University of Campinas)

Abstract

Coordinate descent methods have considerable impact in global optimization because global (or, at least, almost global) minimization is affordable for low-dimensional problems. Coordinate descent methods with high-order regularized models for smooth nonconvex box-constrained minimization are introduced in this work. High-order stationarity asymptotic convergence and first-order stationarity worst-case evaluation complexity bounds are established. The computer work that is necessary for obtaining first-order $$\varepsilon $$ ε -stationarity with respect to the variables of each coordinate-descent block is $$O(\varepsilon ^{-(p+1)/p})$$ O ( ε - ( p + 1 ) / p ) whereas the computer work for getting first-order $$\varepsilon $$ ε -stationarity with respect to all the variables simultaneously is $$O(\varepsilon ^{-(p+1)})$$ O ( ε - ( p + 1 ) ) . Numerical examples involving multidimensional scaling problems are presented. The numerical performance of the methods is enhanced by means of coordinate-descent strategies for choosing initial points.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jglopt:v:84:y:2022:i:3:d:10.1007_s10898-022-01168-6
    DOI: 10.1007/s10898-022-01168-6
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    References listed on IDEAS

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    1. Geovani N. GRAPIGLIA & Yurii NESTEROV, 2017. "Regularized Newton methods for minimizing functions with Hölder continuous Hessians," LIDAM Reprints CORE 2846, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
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    4. NESTEROV, Yurii, 2012. "Efficiency of coordinate descent methods on huge-scale optimization problems," LIDAM Reprints CORE 2511, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Bonettini, Silvia & Prato, Marco & Rebegoldi, Simone, 2016. "A cyclic block coordinate descent method with generalized gradient projections," Applied Mathematics and Computation, Elsevier, vol. 286(C), pages 288-300.
    6. 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.
    7. 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.
    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.
    9. E. G. Birgin & J. M. Martínez, 2019. "A Newton-like method with mixed factorizations and cubic regularization for unconstrained minimization," Computational Optimization and Applications, Springer, vol. 73(3), pages 707-753, July.
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    2. Andrea Brilli & Morteza Kimiaei & Giampaolo Liuzzi & Stefano Lucidi, 2024. "Worst Case Complexity Bounds for Linesearch-Type Derivative-Free Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 419-454, October.

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