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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. 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. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
    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.
    4. 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.
    5. 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.
    6. 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).
    7. 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).
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    9. 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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