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Generalized Proximal Distances for Bilevel Equilibrium Problems

Author

Listed:
  • G. Bento

    (UFG - Universidade Federal de Goiás [Goiânia])

  • J. Cruz Neto

    (UFPI - Universidade Federal do Piauí)

  • J. Lopes

    (UFPI - Universidade Federal do Piauí)

  • A. Soares Jr

    (UFPI - Universidade Federal do Piauí)

  • Antoine Soubeyran

    (GREQAM - Groupement de Recherche en Économie Quantitative d'Aix-Marseille - EHESS - École des hautes études en sciences sociales - AMU - Aix Marseille Université - ECM - École Centrale de Marseille - CNRS - Centre National de la Recherche Scientifique)

Abstract

We consider a bilevel problem involving two monotone equilibrium bifunctions and we show that this problem can be solved by a proximal point method with generalized proximal distances. We propose a framework for the convergence analysis of the sequence generated by the algorithm. This class of problems is very interesting because it covers mathematical programs and optimization problems under equilibrium constraints. As an application, we consider the problem of the stability and change dynamics of a leader-follower relationship in a hierarchical organization.

Suggested Citation

  • G. Bento & J. Cruz Neto & J. Lopes & A. Soares Jr & Antoine Soubeyran, 2016. "Generalized Proximal Distances for Bilevel Equilibrium Problems," Post-Print hal-01690192, HAL.
    • Handle: RePEc:hal:journl:hal-01690192
    DOI: 10.1137/140975589
    Note: View the original document on HAL open archive server: https://amu.hal.science/hal-01690192
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    References listed on IDEAS

    as
    1. 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.
    2. L. Q. Anh & P. Q. Khanh & D. T. M. Van, 2012. "Well-Posedness Under Relaxed Semicontinuity for Bilevel Equilibrium and Optimization Problems with Equilibrium Constraints," Journal of Optimization Theory and Applications, Springer, vol. 153(1), pages 42-59, April.
    3. X. P. Ding, 2010. "Auxiliary Principle and Algorithm for Mixed Equilibrium Problems and Bilevel Mixed Equilibrium Problems in Banach Spaces," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 347-357, August.
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    5. O. Chaldi & Z. Chbani & H. Riahi, 2000. "Equilibrium Problems with Generalized Monotone Bifunctions and Applications to Variational Inequalities," Journal of Optimization Theory and Applications, Springer, vol. 105(2), pages 299-323, May.
    6. Bui Van Dinh & Le Dung Muu, 2011. "On Penalty and Gap Function Methods for Bilevel Equilibrium Problems," Journal of Applied Mathematics, Hindawi, vol. 2011, pages 1-14, November.
    7. Alfredo N. Iusem & B. F. Svaiter & Marc Teboulle, 1994. "Entropy-Like Proximal Methods in Convex Programming," Mathematics of Operations Research, INFORMS, vol. 19(4), pages 790-814, November.
    8. Alfredo N. Iusem & Marc Teboulle, 1995. "Convergence Rate Analysis of Nonquadratic Proximal Methods for Convex and Linear Programming," Mathematics of Operations Research, INFORMS, vol. 20(3), pages 657-677, August.
    9. M. Bianchi & R. Pini, 2005. "Coercivity Conditions for Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 124(1), pages 79-92, January.
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    Cited by:

    1. 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.
    2. 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.
    3. Hideaki Iiduka, 2021. "Inexact stochastic subgradient projection method for stochastic equilibrium problems with nonmonotone bifunctions: application to expected risk minimization in machine learning," Journal of Global Optimization, Springer, vol. 80(2), pages 479-505, June.
    4. Pham Ngoc Anh & Qamrul Hasan Ansari, 2021. "Auxiliary Principle Technique for Hierarchical Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 188(3), pages 882-912, March.

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