IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v87y2024i2d10.1007_s10589-023-00527-7.html
   My bibliography  Save this article

Linearly convergent bilevel optimization with single-step inner methods

Author

Listed:
  • Ensio Suonperä

    (University of Helsinki)

  • Tuomo Valkonen

    (University of Helsinki
    ModeMat, Escuela Politécnica Nacional)

Abstract

We propose a new approach to solving bilevel optimization problems, intermediate between solving full-system optimality conditions with a Newton-type approach, and treating the inner problem as an implicit function. The overall idea is to solve the full-system optimality conditions, but to precondition them to alternate between taking steps of simple conventional methods for the inner problem, the adjoint equation, and the outer problem. While the inner objective has to be smooth, the outer objective may be nonsmooth subject to a prox-contractivity condition. We prove linear convergence of the approach for combinations of gradient descent and forward-backward splitting with exact and inexact solution of the adjoint equation. We demonstrate good performance on learning the regularization parameter for anisotropic total variation image denoising, and the convolution kernel for image deconvolution.

Suggested Citation

  • Ensio Suonperä & Tuomo Valkonen, 2024. "Linearly convergent bilevel optimization with single-step inner methods," Computational Optimization and Applications, Springer, vol. 87(2), pages 571-610, March.
    • Handle: RePEc:spr:coopap:v:87:y:2024:i:2:d:10.1007_s10589-023-00527-7
    DOI: 10.1007/s10589-023-00527-7
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-023-00527-7
    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/s10589-023-00527-7?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. Jörg Fliege & Andrey Tin & Alain Zemkoho, 2021. "Gauss–Newton-type methods for bilevel optimization," Computational Optimization and Applications, Springer, vol. 78(3), pages 793-824, April.
    2. Patrick Mehlitz & Alain B. Zemkoho, 2021. "Sufficient Optimality Conditions in Bilevel Programming," Mathematics of Operations Research, INFORMS, vol. 46(4), pages 1573-1598, November.
    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. Joydeep Dutta & Lahoussine Lafhim & Alain Zemkoho & Shenglong Zhou, 2025. "Nonconvex Quasi-Variational Inequalities: Stability Analysis and Application to Numerical Optimization," Journal of Optimization Theory and Applications, Springer, vol. 204(2), pages 1-43, February.

    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:coopap:v:87:y:2024:i:2:d:10.1007_s10589-023-00527-7. 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.