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Inexact proximal $$\epsilon $$ϵ-subgradient methods for composite convex optimization problems

Author

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  • R. Díaz Millán

    (Federal Institute of Goiás
    IME Federal University of Goiás)

  • M. Pentón Machado

    (Universidade Federal da Bahia)

Abstract

We present two approximate versions of the proximal subgradient method for minimizing the sum of two convex functions (not necessarily differentiable). At each iteration, the algorithms require inexact evaluations of the proximal operator, as well as, approximate subgradients of the functions (namely: the$$\epsilon $$ϵ-subgradients). The methods use different error criteria for approximating the proximal operators. We provide an analysis of the convergence and rate of convergence properties of these methods, considering various stepsize rules, including both, diminishing and constant stepsizes. For the case where one of the functions is smooth, we propose an inexact accelerated version of the proximal gradient method, and prove that the optimal convergence rate for the function values can be achieved. Moreover, we provide some numerical experiments comparing our algorithm with similar recent ones.

Suggested Citation

  • R. Díaz Millán & M. Pentón Machado, 2019. "Inexact proximal $$\epsilon $$ϵ-subgradient methods for composite convex optimization problems," Journal of Global Optimization, Springer, vol. 75(4), pages 1029-1060, December.
    • Handle: RePEc:spr:jglopt:v:75:y:2019:i:4:d:10.1007_s10898-019-00808-8
    DOI: 10.1007/s10898-019-00808-8
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    References listed on IDEAS

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    1. J. Y. Bello Cruz & R. Díaz Millán, 2016. "A relaxed-projection splitting algorithm for variational inequalities in Hilbert spaces," Journal of Global Optimization, Springer, vol. 65(3), pages 597-614, July.
    2. Andrea Simonetto & Hadi Jamali-Rad, 2016. "Primal Recovery from Consensus-Based Dual Decomposition for Distributed Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 168(1), pages 172-197, January.
    3. NESTEROV, Yurii, 2013. "Gradient methods for minimizing composite functions," LIDAM Reprints CORE 2510, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. J. Y. Bello Cruz & R. Díaz Millán, 2014. "A Direct Splitting Method for Nonsmooth Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 161(3), pages 728-737, June.
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    Cited by:

    1. Yunier Bello-Cruz & Max L. N. Gonçalves & Nathan Krislock, 2023. "On FISTA with a relative error rule," Computational Optimization and Applications, Springer, vol. 84(2), pages 295-318, March.

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