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Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations

Author

Listed:
  • Manlio Gaudioso

    (Università della Calabria)

  • Giovanni Giallombardo

    (Università della Calabria)

  • Giovanna Miglionico

    (Università della Calabria)

  • Adil M. Bagirov

    (Federation University Australia)

Abstract

We introduce a proximal bundle method for the numerical minimization of a nonsmooth difference-of-convex (DC) function. Exploiting some classic ideas coming from cutting-plane approaches for the convex case, we iteratively build two separate piecewise-affine approximations of the component functions, grouping the corresponding information in two separate bundles. In the bundle of the first component, only information related to points close to the current iterate are maintained, while the second bundle only refers to a global model of the corresponding component function. We combine the two convex piecewise-affine approximations, and generate a DC piecewise-affine model, which can also be seen as the pointwise maximum of several concave piecewise-affine functions. Such a nonconvex model is locally approximated by means of an auxiliary quadratic program, whose solution is used to certify approximate criticality or to generate a descent search-direction, along with a predicted reduction, that is next explored in a line-search setting. To improve the approximation properties at points that are far from the current iterate a supplementary quadratic program is also introduced to generate an alternative more promising search-direction. We discuss the main convergence issues of the line-search based proximal bundle method, and provide computational results on a set of academic benchmark test problems.

Suggested Citation

  • Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico & Adil M. Bagirov, 2018. "Minimizing nonsmooth DC functions via successive DC piecewise-affine approximations," Journal of Global Optimization, Springer, vol. 71(1), pages 37-55, May.
    • Handle: RePEc:spr:jglopt:v:71:y:2018:i:1:d:10.1007_s10898-017-0568-z
    DOI: 10.1007/s10898-017-0568-z
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    References listed on IDEAS

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    1. R. Horst & N. V. Thoai, 1999. "DC Programming: Overview," Journal of Optimization Theory and Applications, Springer, vol. 103(1), pages 1-43, October.
    2. Le An & Pham Tao, 2005. "The DC (Difference of Convex Functions) Programming and DCA Revisited with DC Models of Real World Nonconvex Optimization Problems," Annals of Operations Research, Springer, vol. 133(1), pages 23-46, January.
    3. Pey-Chun Chen & Pierre Hansen & Brigitte Jaumard & Hoang Tuy, 1998. "Solution of the Multisource Weber and Conditional Weber Problems by D.-C. Programming," Operations Research, INFORMS, vol. 46(4), pages 548-562, August.
    4. Kaisa Joki & Adil M. Bagirov & Napsu Karmitsa & Marko M. Mäkelä, 2017. "A proximal bundle method for nonsmooth DC optimization utilizing nonconvex cutting planes," Journal of Global Optimization, Springer, vol. 68(3), pages 501-535, July.
    5. João Carlos O. Souza & Paulo Roberto Oliveira & Antoine Soubeyran, 2016. "Global convergence of a proximal linearized algorithm for difference of convex functions," Post-Print hal-01440298, HAL.
    6. Demyanov, Alexey V. & Fuduli, Antonio & Miglionico, Giovanna, 2007. "A bundle modification strategy for convex minimization," European Journal of Operational Research, Elsevier, vol. 180(1), pages 38-47, July.
    7. Adil Bagirov & Napsu Karmitsa & Marko M. Mäkelä, 2014. "Introduction to Nonsmooth Optimization," Springer Books, Springer, edition 127, number 978-3-319-08114-4, December.
    8. A. Bagirov & J. Ugon, 2011. "Codifferential method for minimizing nonsmooth DC functions," Journal of Global Optimization, Springer, vol. 50(1), pages 3-22, May.
    9. A. Astorino & A. Fuduli & M. Gaudioso, 2010. "DC models for spherical separation," Journal of Global Optimization, Springer, vol. 48(4), pages 657-669, December.
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    Cited by:

    1. 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.
    2. Chungen Shen & Xiao Liu, 2021. "Solving nonnegative sparsity-constrained optimization via DC quadratic-piecewise-linear approximations," Journal of Global Optimization, Springer, vol. 81(4), pages 1019-1055, December.
    3. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2023. "Sparse optimization via vector k-norm and DC programming with an application to feature selection for support vector machines," Computational Optimization and Applications, Springer, vol. 86(2), pages 745-766, November.
    4. 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.
    5. 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.
    6. M. V. Dolgopolik, 2020. "New global optimality conditions for nonsmooth DC optimization problems," Journal of Global Optimization, Springer, vol. 76(1), pages 25-55, January.
    7. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2022. "Essentials of numerical nonsmooth optimization," Annals of Operations Research, Springer, vol. 314(1), pages 213-253, July.
    8. Welington Oliveira, 2019. "Proximal bundle methods for nonsmooth DC programming," Journal of Global Optimization, Springer, vol. 75(2), pages 523-563, October.
    9. Behzad Pirouz & Behrouz Pirouz, 2023. "Multi-Objective Models for Sparse Optimization in Linear Support Vector Machine Classification," Mathematics, MDPI, vol. 11(17), pages 1-18, August.
    10. Welington Oliveira, 2020. "Sequential Difference-of-Convex Programming," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 936-959, September.
    11. Najmeh Hoseini Monjezi & S. Nobakhtian, 2021. "A filter proximal bundle method for nonsmooth nonconvex constrained optimization," Journal of Global Optimization, Springer, vol. 79(1), pages 1-37, January.
    12. Hoai An Le Thi & Vinh Thanh Ho & Tao Pham Dinh, 2019. "A unified DC programming framework and efficient DCA based approaches for large scale batch reinforcement learning," Journal of Global Optimization, Springer, vol. 73(2), pages 279-310, February.
    13. 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.
    14. Majid E. Abbasov, 2023. "Finding the set of global minimizers of a piecewise affine function," Journal of Global Optimization, Springer, vol. 85(1), pages 1-13, January.
    15. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2020. "Essentials of numerical nonsmooth optimization," 4OR, Springer, vol. 18(1), pages 1-47, March.
    16. Astorino, Annabella & Avolio, Matteo & Fuduli, Antonio, 2022. "A maximum-margin multisphere approach for binary Multiple Instance Learning," European Journal of Operational Research, Elsevier, vol. 299(2), pages 642-652.

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