IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v182y2019i1d10.1007_s10957-019-01500-3.html
   My bibliography  Save this article

Nonsmooth and Nonconvex Optimization via Approximate Difference-of-Convex Decompositions

Author

Listed:
  • Wim Ackooij

    (Électricité de France)

  • Welington Oliveira

    (MINES ParisTech, PSL – Research University, CMA – Centre de Mathématiques Appliquées)

Abstract

We propose an optimization technique for computing stationary points of a broad class of nonsmooth and nonconvex programming problems. The proposed approach (approximately) decomposes the objective function as the difference of two convex functions and performs inexact optimization of the resulting (convex) subproblems. We prove global convergence of our method in the sense that, for an arbitrary starting point, every accumulation point of the sequence of iterates is a Clarke-stationary solution. The given approach is validated by encouraging numerical results on several nonsmooth and nonconvex distributionally robust optimization problems.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:182:y:2019:i:1:d:10.1007_s10957-019-01500-3
    DOI: 10.1007/s10957-019-01500-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-019-01500-3
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-019-01500-3?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Trivedi, Pravin K. & Zimmer, David M., 2007. "Copula Modeling: An Introduction for Practitioners," Foundations and Trends(R) in Econometrics, now publishers, vol. 1(1), pages 1-111, April.
    2. Strekalovsky, Alexander S., 2015. "On local search in d.c. optimization problems," Applied Mathematics and Computation, Elsevier, vol. 255(C), pages 73-83.
    3. Krzysztof Czesław Kiwiel, 1985. "A Linearization Algorithm for Nonsmooth Minimization," Mathematics of Operations Research, INFORMS, vol. 10(2), pages 185-194, May.
    4. Wim Ackooij & Welington Oliveira, 2014. "Level bundle methods for constrained convex optimization with various oracles," Computational Optimization and Applications, Springer, vol. 57(3), pages 555-597, April.
    5. Welington Oliveira, 2019. "Proximal bundle methods for nonsmooth DC programming," Journal of Global Optimization, Springer, vol. 75(2), pages 523-563, October.
    6. Minh N. Dao & Joachim Gwinner & Dominikus Noll & Nina Ovcharova, 2016. "Nonconvex bundle method with application to a delamination problem," Computational Optimization and Applications, Springer, vol. 65(1), pages 173-203, September.
    7. J. Bello Cruz & W. Oliveira, 2014. "Level bundle-like algorithms for convex optimization," Journal of Global Optimization, Springer, vol. 59(4), pages 787-809, August.
    8. 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.
    9. W. Hare & C. Sagastizábal & M. Solodov, 2016. "A proximal bundle method for nonsmooth nonconvex functions with inexact information," Computational Optimization and Applications, Springer, vol. 63(1), pages 1-28, January.
    10. 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.
    11. Jong-Shi Pang & Meisam Razaviyayn & Alberth Alvarado, 2017. "Computing B-Stationary Points of Nonsmooth DC Programs," Mathematics of Operations Research, INFORMS, vol. 42(1), pages 95-118, January.
    12. Lemaréchal, C. & Nemirovskii, A. & Nesterov, Y., 1995. "New variants of bundle methods," LIDAM Reprints CORE 1166, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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. Welington Oliveira, 2020. "Sequential Difference-of-Convex Programming," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 936-959, September.
    3. 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.
    4. 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.
    5. 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.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Welington Oliveira, 2019. "Proximal bundle methods for nonsmooth DC programming," Journal of Global Optimization, Springer, vol. 75(2), pages 523-563, October.
    2. 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.
    3. Najmeh Hoseini Monjezi & S. Nobakhtian, 2019. "A new infeasible proximal bundle algorithm for nonsmooth nonconvex constrained optimization," Computational Optimization and Applications, Springer, vol. 74(2), pages 443-480, November.
    4. Welington Oliveira, 2020. "Sequential Difference-of-Convex Programming," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 936-959, September.
    5. 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.
    6. Manlio Gaudioso & Giovanni Giallombardo & Giovanna Miglionico, 2020. "Essentials of numerical nonsmooth optimization," 4OR, Springer, vol. 18(1), pages 1-47, March.
    7. Yunmei Chen & Xiaojing Ye & Wei Zhang, 2020. "Acceleration techniques for level bundle methods in weakly smooth convex constrained optimization," Computational Optimization and Applications, Springer, vol. 77(2), pages 411-432, November.
    8. 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.
    9. Xiaoliang Wang & Liping Pang & Qi Wu & Mingkun Zhang, 2021. "An Adaptive Proximal Bundle Method with Inexact Oracles for a Class of Nonconvex and Nonsmooth Composite Optimization," Mathematics, MDPI, vol. 9(8), pages 1-27, April.
    10. Tang, Chunming & Liu, Shuai & Jian, Jinbao & Ou, Xiaomei, 2020. "A multi-step doubly stabilized bundle method for nonsmooth convex optimization," Applied Mathematics and Computation, Elsevier, vol. 376(C).
    11. 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.
    12. 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.
    13. 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.
    14. Wim van Ackooij & Welington de Oliveira & Yongjia Song, 2018. "Adaptive Partition-Based Level Decomposition Methods for Solving Two-Stage Stochastic Programs with Fixed Recourse," INFORMS Journal on Computing, INFORMS, vol. 30(1), pages 57-70, February.
    15. 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.
    16. Wim Ackooij & Welington Oliveira & Yongjia Song, 2019. "On level regularization with normal solutions in decomposition methods for multistage stochastic programming problems," Computational Optimization and Applications, Springer, vol. 74(1), pages 1-42, September.
    17. Shuai Liu, 2019. "A simple version of bundle method with linear programming," Computational Optimization and Applications, Springer, vol. 72(2), pages 391-412, March.
    18. Jian Lv & Li-Ping Pang & Fan-Yun Meng, 2018. "A proximal bundle method for constrained nonsmooth nonconvex optimization with inexact information," Journal of Global Optimization, Springer, vol. 70(3), pages 517-549, March.
    19. Blanchot, Xavier & Clautiaux, François & Detienne, Boris & Froger, Aurélien & Ruiz, Manuel, 2023. "The Benders by batch algorithm: Design and stabilization of an enhanced algorithm to solve multicut Benders reformulation of two-stage stochastic programs," European Journal of Operational Research, Elsevier, vol. 309(1), pages 202-216.
    20. 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).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:182:y:2019:i:1:d:10.1007_s10957-019-01500-3. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.