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Proximal bundle methods for nonsmooth DC programming

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  • Welington Oliveira

    (CMA – Centre de Mathématiques Appliquées)

Abstract

We consider the problem of minimizing the difference of two nonsmooth convex functions over a simple convex set. To deal with this class of nonsmooth and nonconvex optimization problems, we propose new proximal bundle algorithms and show that the given approaches generate subsequences of iterates that converge to critical points. Trial points are obtained by solving strictly convex master programs defined by the sum of a convex cutting-plane model and a freely-chosen Bregman function. In the unconstrained case with the Bregman function being the Euclidean distance, new iterates are solutions of strictly convex quadratic programs of limited sizes. Stronger convergence results (d-stationarity) can be achieved depending on (a) further assumptions on the second DC component of the objective function and (b) solving possibly more than one master program at certain iterations. The given approaches are validated by encouraging numerical results on some academic DC programs.

Suggested Citation

  • Welington Oliveira, 2019. "Proximal bundle methods for nonsmooth DC programming," Journal of Global Optimization, Springer, vol. 75(2), pages 523-563, October.
    • Handle: RePEc:spr:jglopt:v:75:y:2019:i:2:d:10.1007_s10898-019-00755-4
    DOI: 10.1007/s10898-019-00755-4
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    References listed on IDEAS

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    Cited by:

    1. Pietro D’Alessandro & Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2024. "The Descent–Ascent Algorithm for DC Programming," INFORMS Journal on Computing, INFORMS, vol. 36(2), pages 657-671, March.
    2. Byun, Ji-Eun & de Oliveira, Welington & Royset, Johannes O., 2023. "S-BORM: Reliability-based optimization of general systems using buffered optimization and reliability method," Reliability Engineering and System Safety, Elsevier, vol. 236(C).
    3. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2022. "Essentials of numerical nonsmooth optimization," Annals of Operations Research, Springer, vol. 314(1), pages 213-253, July.
    4. M. V. Dolgopolik, 2022. "DC Semidefinite programming and cone constrained DC optimization I: theory," Computational Optimization and Applications, Springer, vol. 82(3), pages 649-671, July.
    5. W. Ackooij & S. Demassey & P. Javal & H. Morais & W. Oliveira & B. Swaminathan, 2021. "A bundle method for nonsmooth DC programming with application to chance-constrained problems," Computational Optimization and Applications, Springer, vol. 78(2), pages 451-490, March.
    6. Wim Ackooij & Welington Oliveira, 2019. "Nonsmooth and Nonconvex Optimization via Approximate Difference-of-Convex Decompositions," Journal of Optimization Theory and Applications, Springer, vol. 182(1), pages 49-80, July.
    7. Welington Oliveira, 2020. "Sequential Difference-of-Convex Programming," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 936-959, September.
    8. M. V. Dolgopolik, 2023. "DC semidefinite programming and cone constrained DC optimization II: local search methods," Computational Optimization and Applications, Springer, vol. 85(3), pages 993-1031, July.
    9. A. M. Bagirov & N. Hoseini Monjezi & S. Taheri, 2021. "An augmented subgradient method for minimizing nonsmooth DC functions," Computational Optimization and Applications, Springer, vol. 80(2), pages 411-438, November.
    10. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2020. "Essentials of numerical nonsmooth optimization," 4OR, Springer, vol. 18(1), pages 1-47, March.
    11. Butyn, Emerson & Karas, Elizabeth W. & de Oliveira, Welington, 2022. "A derivative-free trust-region algorithm with copula-based models for probability maximization problems," European Journal of Operational Research, Elsevier, vol. 298(1), pages 59-75.

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