IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v142y2009i1d10.1007_s10957-009-9529-0.html
   My bibliography  Save this article

Regularization Algorithms for Solving Monotone Ky Fan Inequalities with Application to a Nash-Cournot Equilibrium Model

Author

Listed:
  • L. D. Muu

    (VAST)

  • T. D. Quoc

    (Hanoi University of Science)

Abstract

We make use of the Banach contraction mapping principle to prove the linear convergence of a regularization algorithm for strongly monotone Ky Fan inequalities that satisfy a Lipschitz-type condition recently introduced by Mastroeni. We then modify the proposed algorithm to obtain a line search-free algorithm which does not require the Lipschitz-type condition. We apply the proposed algorithms to implement inexact proximal methods for solving monotone (not necessarily strongly monotone) Ky Fan inequalities. Applications to variational inequality and complementarity problems are discussed. As a consequence, a linearly convergent derivative-free algorithm without line search for strongly monotone nonlinear complementarity problem is obtained. Application to a Nash-Cournot equilibrium model is discussed and some preliminary computational results are reported.

Suggested Citation

  • L. D. Muu & T. D. Quoc, 2009. "Regularization Algorithms for Solving Monotone Ky Fan Inequalities with Application to a Nash-Cournot Equilibrium Model," Journal of Optimization Theory and Applications, Springer, vol. 142(1), pages 185-204, July.
    • Handle: RePEc:spr:joptap:v:142:y:2009:i:1:d:10.1007_s10957-009-9529-0
    DOI: 10.1007/s10957-009-9529-0
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-009-9529-0
    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/s10957-009-9529-0?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. I.V. Konnov, 2003. "Application of the Proximal Point Method to Nonmonotone Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 119(2), pages 317-333, November.
    2. M. A. Noor, 2004. "Auxiliary Principle Technique for Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 122(2), pages 371-386, August.
    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. Tran Quoc & Le Muu, 2012. "Iterative methods for solving monotone equilibrium problems via dual gap functions," Computational Optimization and Applications, Springer, vol. 51(2), pages 709-728, March.
    2. Phan Tu Vuong & Jean Jacques Strodiot, 2018. "The Glowinski–Le Tallec splitting method revisited in the framework of equilibrium problems in Hilbert spaces," Journal of Global Optimization, Springer, vol. 70(2), pages 477-495, February.
    3. Stefano Lucidi & Mauro Passacantando & Francesco Rinaldi, 2022. "Solving non-monotone equilibrium problems via a DIRECT-type approach," Journal of Global Optimization, Springer, vol. 83(4), pages 699-725, August.
    4. Yekini Shehu & Lulu Liu & Xiaolong Qin & Qiao-Li Dong, 2022. "Reflected Iterative Method for Non-Monotone Equilibrium Problems with Applications to Nash-Cournot Equilibrium Models," Networks and Spatial Economics, Springer, vol. 22(1), pages 153-180, March.
    5. Pasakorn Yordsorn & Poom Kumam & Habib ur Rehman & Abdulkarim Hassan Ibrahim, 2020. "A Weak Convergence Self-Adaptive Method for Solving Pseudomonotone Equilibrium Problems in a Real Hilbert Space," Mathematics, MDPI, vol. 8(7), pages 1-24, July.
    6. Jean Strodiot & Phan Vuong & Thi Nguyen, 2016. "A class of shrinking projection extragradient methods for solving non-monotone equilibrium problems in Hilbert spaces," Journal of Global Optimization, Springer, vol. 64(1), pages 159-178, January.
    7. Bigi, Giancarlo & Castellani, Marco & Pappalardo, Massimo & Passacantando, Mauro, 2013. "Existence and solution methods for equilibria," European Journal of Operational Research, Elsevier, vol. 227(1), pages 1-11.
    8. Habib ur Rehman & Poom Kumam & Meshal Shutaywi & Nasser Aedh Alreshidi & Wiyada Kumam, 2020. "Inertial Optimization Based Two-Step Methods for Solving Equilibrium Problems with Applications in Variational Inequality Problems and Growth Control Equilibrium Models," Energies, MDPI, vol. 13(12), pages 1-28, June.
    9. Dang Hieu, 2018. "An inertial-like proximal algorithm for equilibrium problems," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 88(3), pages 399-415, December.
    10. Le Quang Thuy & Trinh Ngoc Hai, 2017. "A Projected Subgradient Algorithm for Bilevel Equilibrium Problems and Applications," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 411-431, November.
    11. Phan Vuong & Jean Strodiot & Van Nguyen, 2014. "Projected viscosity subgradient methods for variational inequalities with equilibrium problem constraints in Hilbert spaces," Journal of Global Optimization, Springer, vol. 59(1), pages 173-190, May.
    12. Mostafa Nasri & Luiz Carlos Matioli & Euda Mara Silva Ferreira & Adilson Silveira, 2016. "Implementation of Augmented Lagrangian Methods for Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 971-991, March.
    13. Habib ur Rehman & Poom Kumam & Ioannis K. Argyros & Meshal Shutaywi & Zahir Shah, 2020. "Optimization Based Methods for Solving the Equilibrium Problems with Applications in Variational Inequality Problems and Solution of Nash Equilibrium Models," Mathematics, MDPI, vol. 8(5), pages 1-28, May.
    14. Phan Tu Vuong & Jean Jacques Strodiot, 2020. "A Dynamical System for Strongly Pseudo-monotone Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 185(3), pages 767-784, June.
    15. Pham Ky Anh & Trinh Ngoc Hai, 2019. "Novel self-adaptive algorithms for non-Lipschitz equilibrium problems with applications," Journal of Global Optimization, Springer, vol. 73(3), pages 637-657, March.
    16. Lateef Olakunle Jolaoso & Christian Chibueze Okeke & Yekini Shehu, 2021. "Extragradient Algorithm for Solving Pseudomonotone Equilibrium Problem with Bregman Distance in Reflexive Banach Spaces," Networks and Spatial Economics, Springer, vol. 21(4), pages 873-903, 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. P. Anh & T. Hai & P. Tuan, 2016. "On ergodic algorithms for equilibrium problems," Journal of Global Optimization, Springer, vol. 64(1), pages 179-195, January.
    2. Nils Langenberg, 2012. "Interior point methods for equilibrium problems," Computational Optimization and Applications, Springer, vol. 53(2), pages 453-483, October.
    3. Tran Quoc & Le Muu, 2012. "Iterative methods for solving monotone equilibrium problems via dual gap functions," Computational Optimization and Applications, Springer, vol. 51(2), pages 709-728, March.
    4. N. N. Tam & J. C. Yao & N. D. Yen, 2008. "Solution Methods for Pseudomonotone Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 138(2), pages 253-273, August.
    5. Pasakorn Yordsorn & Poom Kumam & Habib ur Rehman & Abdulkarim Hassan Ibrahim, 2020. "A Weak Convergence Self-Adaptive Method for Solving Pseudomonotone Equilibrium Problems in a Real Hilbert Space," Mathematics, MDPI, vol. 8(7), pages 1-24, July.
    6. P. Anh & H. Le Thi, 2013. "An Armijo-type method for pseudomonotone equilibrium problems and its applications," Journal of Global Optimization, Springer, vol. 57(3), pages 803-820, November.
    7. J. X. Cruz Neto & F. M. O. Jacinto & P. A. Soares & J. C. O. Souza, 2018. "On maximal monotonicity of bifunctions on Hadamard manifolds," Journal of Global Optimization, Springer, vol. 72(3), pages 591-601, November.
    8. A. Iusem & F. Lara, 2022. "Proximal Point Algorithms for Quasiconvex Pseudomonotone Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 193(1), pages 443-461, June.
    9. M. A. Noor & K. I. Noor & E. Al-Said, 2011. "Auxiliary Principle Technique for Solving Bifunction Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 149(2), pages 441-445, May.
    10. J. X. Cruz Neto & J. O. Lopes & A. Soubeyran & J. C. O. Souza, 2022. "Abstract regularized equilibria: application to Becker’s household behavior theory," Annals of Operations Research, Springer, vol. 316(2), pages 1279-1300, September.
    11. Ke Han & Terry L. Friesz, 2017. "Continuity of the Effective Delay Operator for Networks Based on the Link Delay Model," Networks and Spatial Economics, Springer, vol. 17(4), pages 1095-1110, December.
    12. Alfredo Iusem & Felipe Lara & Raúl T. Marcavillaca & Le Hai Yen, 2024. "A Two-Step Proximal Point Algorithm for Nonconvex Equilibrium Problems with Applications to Fractional Programming," Journal of Global Optimization, Springer, vol. 90(3), pages 755-779, November.
    13. Le Quang Thuy & Trinh Ngoc Hai, 2017. "A Projected Subgradient Algorithm for Bilevel Equilibrium Problems and Applications," Journal of Optimization Theory and Applications, Springer, vol. 175(2), pages 411-431, November.
    14. Habib ur Rehman & Poom Kumam & Meshal Shutaywi & Nasser Aedh Alreshidi & Wiyada Kumam, 2020. "Inertial Optimization Based Two-Step Methods for Solving Equilibrium Problems with Applications in Variational Inequality Problems and Growth Control Equilibrium Models," Energies, MDPI, vol. 13(12), pages 1-28, June.
    15. Han, Ke & Szeto, W.Y. & Friesz, Terry L., 2015. "Formulation, existence, and computation of boundedly rational dynamic user equilibrium with fixed or endogenous user tolerance," Transportation Research Part B: Methodological, Elsevier, vol. 79(C), pages 16-49.
    16. Bigi, Giancarlo & Castellani, Marco & Pappalardo, Massimo & Passacantando, Mauro, 2013. "Existence and solution methods for equilibria," European Journal of Operational Research, Elsevier, vol. 227(1), pages 1-11.
    17. Han, Ke & Friesz, Terry L. & Szeto, W.Y. & Liu, Hongcheng, 2015. "Elastic demand dynamic network user equilibrium: Formulation, existence and computation," Transportation Research Part B: Methodological, Elsevier, vol. 81(P1), pages 183-209.
    18. Abdellatif Moudafi, 2010. "Proximal methods for a class of bilevel monotone equilibrium problems," Journal of Global Optimization, Springer, vol. 47(2), pages 287-292, June.
    19. Song, Wenjing & Han, Ke & Wang, Yiou & Friesz, Terry L. & del Castillo, Enrique, 2018. "Statistical metamodeling of dynamic network loading," Transportation Research Part B: Methodological, Elsevier, vol. 117(PB), pages 740-756.
    20. E. Allevi & A. Gnudi & I. Konnov, 2006. "The Proximal Point Method for Nonmonotone Variational Inequalities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 63(3), pages 553-565, July.

    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:142:y:2009:i:1:d:10.1007_s10957-009-9529-0. 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.