IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v155y2012i3d10.1007_s10957-012-0111-9.html
   My bibliography  Save this article

An Interior Proximal Method for a Class of Quasimonotone Variational Inequalities

Author

Listed:
  • Nils Langenberg

    (University of Trier)

Abstract

The Bregman-function-based Proximal Point Algorithm for variational inequalities is studied. Classical papers on this method deal with the assumption that the operator of the variational inequality is monotone. Motivated by the fact that this assumption can be considered to be restrictive, e.g., in the discussion of Nash equilibrium problems, the main objective of the present paper is to provide a convergence analysis only using a weaker assumption called quasimonotonicity. To the best of our knowledge, this is the first algorithm established for this general and frequently studied class of problems.

Suggested Citation

  • Nils Langenberg, 2012. "An Interior Proximal Method for a Class of Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 155(3), pages 902-922, December.
    • Handle: RePEc:spr:joptap:v:155:y:2012:i:3:d:10.1007_s10957-012-0111-9
    DOI: 10.1007/s10957-012-0111-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-012-0111-9
    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/s10957-012-0111-9?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. D. Aussel & N. Hadjisavvas, 2004. "On Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 121(2), pages 445-450, May.
    2. Nils Langenberg & Rainer Tichatschke, 2012. "Interior proximal methods for quasiconvex optimization," Journal of Global Optimization, Springer, vol. 52(3), pages 641-661, March.
    3. M. V. Solodov & B. F. Svaiter, 2000. "An Inexact Hybrid Generalized Proximal Point Algorithm and Some New Results on the Theory of Bregman Functions," Mathematics of Operations Research, INFORMS, vol. 25(2), pages 214-230, May.
    4. Nils Langenberg, 2010. "Pseudomonotone operators and the Bregman Proximal Point Algorithm," Journal of Global Optimization, Springer, vol. 47(4), pages 537-555, August.
    5. I. V. Konnov, 1998. "On Quasimonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 99(1), pages 165-181, October.
    6. Jonathan Eckstein, 1993. "Nonlinear Proximal Point Algorithms Using Bregman Functions, with Applications to Convex Programming," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 202-226, February.
    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. Minglu Ye & Yiran He, 2015. "A double projection method for solving variational inequalities without monotonicity," Computational Optimization and Applications, Springer, vol. 60(1), pages 141-150, January.
    2. Massimiliano Giuli, 2013. "Closedness of the Solution Map in Quasivariational Inequalities of Ky Fan Type," Journal of Optimization Theory and Applications, Springer, vol. 158(1), pages 130-144, July.
    3. Nils Langenberg, 2015. "Interior Proximal Method Without the Cutting Plane Property," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 529-557, August.
    4. Xin He & Nan-jing Huang & Xue-song Li, 2022. "Modified Projection Methods for Solving Multi-valued Variational Inequality without Monotonicity," Networks and Spatial Economics, Springer, vol. 22(2), pages 361-377, June.
    5. Hongwei Liu & Jun Yang, 2020. "Weak convergence of iterative methods for solving quasimonotone variational inequalities," Computational Optimization and Applications, Springer, vol. 77(2), pages 491-508, November.

    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. Nils Langenberg, 2015. "Interior Proximal Method Without the Cutting Plane Property," Journal of Optimization Theory and Applications, Springer, vol. 166(2), pages 529-557, August.
    2. R. P. Agarwal & M. Balaj & D. O’Regan, 2014. "A Common Fixed Point Theorem with Applications," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 482-490, November.
    3. Jonathan Eckstein & Paulo Silva, 2010. "Proximal methods for nonlinear programming: double regularization and inexact subproblems," Computational Optimization and Applications, Springer, vol. 46(2), pages 279-304, June.
    4. Nils Langenberg & Rainer Tichatschke, 2012. "Interior proximal methods for quasiconvex optimization," Journal of Global Optimization, Springer, vol. 52(3), pages 641-661, March.
    5. Papa Quiroz, E.A. & Mallma Ramirez, L. & Oliveira, P.R., 2015. "An inexact proximal method for quasiconvex minimization," European Journal of Operational Research, Elsevier, vol. 246(3), pages 721-729.
    6. Nils Langenberg, 2012. "Interior point methods for equilibrium problems," Computational Optimization and Applications, Springer, vol. 53(2), pages 453-483, October.
    7. Souza, Sissy da S. & Oliveira, P.R. & da Cruz Neto, J.X. & Soubeyran, A., 2010. "A proximal method with separable Bregman distances for quasiconvex minimization over the nonnegative orthant," European Journal of Operational Research, Elsevier, vol. 201(2), pages 365-376, March.
    8. Yiran He, 2017. "Solvability of the Minty Variational Inequality," Journal of Optimization Theory and Applications, Springer, vol. 174(3), pages 686-692, September.
    9. E. A. Papa Quiroz & S. Cruzado, 2022. "An inexact scalarization proximal point method for multiobjective quasiconvex minimization," Annals of Operations Research, Springer, vol. 316(2), pages 1445-1470, September.
    10. Xin Jiang & Lieven Vandenberghe, 2023. "Bregman Three-Operator Splitting Methods," Journal of Optimization Theory and Applications, Springer, vol. 196(3), pages 936-972, March.
    11. Xin Jiang & Lieven Vandenberghe, 2022. "Bregman primal–dual first-order method and application to sparse semidefinite programming," Computational Optimization and Applications, Springer, vol. 81(1), pages 127-159, January.
    12. J. X. Cruz Neto & O. P. Ferreira & P. R. Oliveira & R. C. M. Silva, 2008. "Central Paths in Semidefinite Programming, Generalized Proximal-Point Method and Cauchy Trajectories in Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 139(2), pages 227-242, November.
    13. Sorin-Mihai Grad & Felipe Lara, 2022. "An extension of the proximal point algorithm beyond convexity," Journal of Global Optimization, Springer, vol. 82(2), pages 313-329, February.
    14. M. Bianchi & R. Pini, 2005. "Coercivity Conditions for Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 124(1), pages 79-92, January.
    15. Bingsheng He & Li-Zhi Liao & Xiang Wang, 2012. "Proximal-like contraction methods for monotone variational inequalities in a unified framework I: Effective quadruplet and primary methods," Computational Optimization and Applications, Springer, vol. 51(2), pages 649-679, March.
    16. J. X. Cruz Neto & P. R. Oliveira & A. Soubeyran & J. C. O. Souza, 2020. "A generalized proximal linearized algorithm for DC functions with application to the optimal size of the firm problem," Annals of Operations Research, Springer, vol. 289(2), pages 313-339, June.
    17. William W. Hager & Hongchao Zhang, 2020. "Convergence rates for an inexact ADMM applied to separable convex optimization," Computational Optimization and Applications, Springer, vol. 77(3), pages 729-754, December.
    18. Papa Quiroz, E.A. & Roberto Oliveira, P., 2012. "An extension of proximal methods for quasiconvex minimization on the nonnegative orthant," European Journal of Operational Research, Elsevier, vol. 216(1), pages 26-32.
    19. G. Isac & S. Z. Németh, 2008. "REFE-Acceptable Mappings: Necessary and Sufficient Condition for the Nonexistence of a Regular Exceptional Family of Elements," Journal of Optimization Theory and Applications, Springer, vol. 137(3), pages 507-520, June.
    20. Regina Sandra Burachik & B. F. Svaiter, 2001. "A Relative Error Tolerance for a Family of Generalized Proximal Point Methods," Mathematics of Operations Research, INFORMS, vol. 26(4), pages 816-831, November.

    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:155:y:2012:i:3:d:10.1007_s10957-012-0111-9. 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.