IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v171y2016i3d10.1007_s10957-015-0813-x.html
   My bibliography  Save this article

Maximal Monotone Inclusions and Fitzpatrick Functions

Author

Listed:
  • J. M. Borwein

    (University of Newcastle)

  • J. Dutta

    (Indian Institute of Technology Kanpur)

Abstract

In this paper, we study maximal monotone inclusions from the perspective of gap functions. We propose a very natural gap function for an arbitrary maximal monotone inclusion and will demonstrate how naturally this gap function arises from the Fitzpatrick function, which is a convex function, used to represent maximal monotone operators. This allows us to use the powerful strong Fitzpatrick inequality to analyse solutions of the inclusion. We also study the special cases of a variational inequality and of a generalized variational inequality problem. The associated notion of a scalar gap is also considered in some detail. Corresponding local and global error bounds are also developed for the maximal monotone inclusion.

Suggested Citation

  • J. M. Borwein & J. Dutta, 2016. "Maximal Monotone Inclusions and Fitzpatrick Functions," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 757-784, December.
    • Handle: RePEc:spr:joptap:v:171:y:2016:i:3:d:10.1007_s10957-015-0813-x
    DOI: 10.1007/s10957-015-0813-x
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-015-0813-x
    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/s10957-015-0813-x?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. NESTEROV, Yurii & SCRIMALI, Laura, 2011. "Solving strongly monotone variational and quasi-variational inequalities," LIDAM Reprints CORE 2357, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. D. Aussel & J. Dutta, 2011. "On Gap Functions for Multivalued Stampacchia Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 149(3), pages 513-527, June.
    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. Shisheng Cui & Uday Shanbhag & Mathias Staudigl & Phan Vuong, 2022. "Stochastic relaxed inertial forward-backward-forward splitting for monotone inclusions in Hilbert spaces," Computational Optimization and Applications, Springer, vol. 83(2), pages 465-524, November.
    2. Panayotis Mertikopoulos & Mathias Staudigl, 2018. "Stochastic Mirror Descent Dynamics and Their Convergence in Monotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 838-867, December.

    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. Eleftherios Couzoudis & Philipp Renner, 2013. "Computing generalized Nash equilibria by polynomial programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 77(3), pages 459-472, June.
    2. Fedor Stonyakin & Alexander Gasnikov & Pavel Dvurechensky & Alexander Titov & Mohammad Alkousa, 2022. "Generalized Mirror Prox Algorithm for Monotone Variational Inequalities: Universality and Inexact Oracle," Journal of Optimization Theory and Applications, Springer, vol. 194(3), pages 988-1013, September.
    3. Giancarlo Bigi & Mauro Passacantando, 2016. "Gap functions for quasi-equilibria," Journal of Global Optimization, Springer, vol. 66(4), pages 791-810, December.
    4. Maicon Marques Alves & B. F. Svaiter, 2016. "A Variant of the Hybrid Proximal Extragradient Method for Solving Strongly Monotone Inclusions and its Complexity Analysis," Journal of Optimization Theory and Applications, Springer, vol. 168(1), pages 198-215, January.
    5. Laura Scrimali, 2012. "Infinite Dimensional Duality Theory Applied to Investment Strategies in Environmental Policy," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 258-277, July.
    6. Samir Adly & Monica G. Cojocaru & Ba Khiet Le, 2024. "State-Dependent Sweeping Processes: Asymptotic Behavior and Algorithmic Approaches," Journal of Optimization Theory and Applications, Springer, vol. 202(2), pages 932-948, August.
    7. Denizalp Goktas & Amy Greenwald, 2022. "Gradient Descent Ascent in Min-Max Stackelberg Games," Papers 2208.09690, arXiv.org.
    8. Khan, Suhel Ahmad & Chen, Jia-wei, 2015. "Gap function and global error bounds for generalized mixed quasi variational inequalities," Applied Mathematics and Computation, Elsevier, vol. 260(C), pages 71-81.
    9. Yekini Shehu & Aviv Gibali & Simone Sagratella, 2020. "Inertial Projection-Type Methods for Solving Quasi-Variational Inequalities in Real Hilbert Spaces," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 877-894, March.
    10. Denizalp Goktas & Jiayi Zhao & Amy Greenwald, 2022. "Robust No-Regret Learning in Min-Max Stackelberg Games," Papers 2203.14126, arXiv.org, revised Apr 2022.
    11. D. Aussel & R. Correa & M. Marechal, 2011. "Gap Functions for Quasivariational Inequalities and Generalized Nash Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 151(3), pages 474-488, December.
    12. Xiao-bo Li & Li-wen Zhou & Nan-jing Huang, 2016. "Gap Functions and Global Error Bounds for Generalized Mixed Variational Inequalities on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 830-849, March.

    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:171:y:2016:i:3:d:10.1007_s10957-015-0813-x. 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.