IDEAS home Printed from https://ideas.repec.org/a/inm/ormoor/v49y2024i4p2385-2401.html

The Cost of Nonconvexity in Deterministic Nonsmooth Optimization

Author

Listed:
  • Siyu Kong

    (School of Operations Research and Information Engineering, Cornell University, Ithaca, New York 14853)

  • A. S. Lewis

    (School of Operations Research and Information Engineering, Cornell University, Ithaca, New York 14853)

Abstract

We study the impact of nonconvexity on the complexity of nonsmooth optimization, emphasizing objectives such as piecewise linear functions, which may not be weakly convex. We focus on a dimension-independent analysis, slightly modifying a 2020 black-box algorithm of Zhang-Lin-Jegelka-Sra-Jadbabaie that approximates an ϵ -stationary point of any directionally differentiable Lipschitz objective using O ( ϵ − 4 ) calls to a specialized subgradient oracle and a randomized line search. Seeking by contrast a deterministic method, we present a simple black-box version that achieves O ( ϵ − 5 ) for any difference-of-convex objective and O ( ϵ − 4 ) for the weakly convex case. Our complexity bound depends on a natural nonconvexity modulus that is related, intriguingly, to the negative part of directional second derivatives of the objective, understood in the distributional sense.

Suggested Citation

  • Siyu Kong & A. S. Lewis, 2024. "The Cost of Nonconvexity in Deterministic Nonsmooth Optimization," Mathematics of Operations Research, INFORMS, vol. 49(4), pages 2385-2401, November.
    • Handle: RePEc:inm:ormoor:v:49:y:2024:i:4:p:2385-2401
    DOI: 10.1287/moor.2022.0289
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/moor.2022.0289
    Download Restriction: no
    File URL: https://libkey.io/10.1287/moor.2022.0289?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
    ---><---

    References listed on IDEAS

    as
    1. Robert Mifflin, 1977. "An Algorithm for Constrained Optimization with Semismooth Functions," Mathematics of Operations Research, INFORMS, vol. 2(2), pages 191-207, May.
    2. Nezam Mahdavi-Amiri & Rohollah Yousefpour, 2012. "An Effective Nonsmooth Optimization Algorithm for Locally Lipschitz Functions," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 180-195, October.
    Full references (including those not matched with items on IDEAS)

    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. Bennet Gebken, 2024. "A note on the convergence of deterministic gradient sampling in nonsmooth optimization," Computational Optimization and Applications, Springer, vol. 88(1), pages 151-165, May.
    2. Bennet Gebken, 2025. "Analyzing the Speed of Convergence in Nonsmooth Optimization via the Goldstein subdifferential," Journal of Optimization Theory and Applications, Springer, vol. 206(3), pages 1-38, September.
    3. Smail Addoune & Karima Boufi & Ahmed Roubi, 2018. "Proximal Bundle Algorithms for Nonlinearly Constrained Convex Minimax Fractional Programs," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 212-239, October.
    4. Churlzu Lim & Hanif Sherali, 2006. "A Trust Region Target Value Method for Optimizing Nondifferentiable Lagrangian Duals of Linear Programs," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 64(1), pages 33-53, August.
    5. Z. Akbari & R. Yousefpour & M. Reza Peyghami, 2015. "A New Nonsmooth Trust Region Algorithm for Locally Lipschitz Unconstrained Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 164(3), pages 733-754, March.
    6. G. Di Pillo & L. Grippo & S. Lucidi, 1997. "Smooth Transformation of the Generalized Minimax Problem," Journal of Optimization Theory and Applications, Springer, vol. 95(1), pages 1-24, October.
    7. H. Boualam & A. Roubi, 2019. "Proximal bundle methods based on approximate subgradients for solving Lagrangian duals of minimax fractional programs," Journal of Global Optimization, Springer, vol. 74(2), pages 255-284, June.
    8. 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.
    9. K. Kiwiel, 1994. "A Bundle of Method for Minimizing a Sum of Convex Functions with Smooth Weights," Working Papers wp94013, International Institute for Applied Systems Analysis.
    10. N. Mahdavi-Amiri & M. Shaeiri, 2020. "A conjugate gradient sampling method for nonsmooth optimization," 4OR, Springer, vol. 18(1), pages 73-90, March.
    11. Hanif D. Sherali & Churlzu Lim, 2007. "Enhancing Lagrangian Dual Optimization for Linear Programs by Obviating Nondifferentiability," INFORMS Journal on Computing, INFORMS, vol. 19(1), pages 3-13, February.
    12. A. M. Bagirov & B. Karasözen & M. Sezer, 2008. "Discrete Gradient Method: Derivative-Free Method for Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 137(2), pages 317-334, May.
    13. Nezam Mahdavi-Amiri & Rohollah Yousefpour, 2012. "An Effective Nonsmooth Optimization Algorithm for Locally Lipschitz Functions," Journal of Optimization Theory and Applications, Springer, vol. 155(1), pages 180-195, October.
    14. Bennet Gebken & Sebastian Peitz, 2021. "An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 696-723, March.
    15. Maleknia, Morteza & Soleimani-damaneh, Majid, 2024. "An effective subgradient algorithm via Mifflin’s line search for nonsmooth nonconvex multiobjective optimization," European Journal of Operational Research, Elsevier, vol. 319(2), pages 505-516.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;
    ;
    ;
    ;

    JEL classification:

    Statistics

    Access and download statistics

    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:inm:ormoor:v:49:y:2024:i:4:p:2385-2401. 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: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    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.