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Application of the Proximal Point Method to Nonmonotone Equilibrium Problems

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  • I.V. Konnov

    (Kazan University)

Abstract

We consider a general equilibrium problem defined on a convex set, whose cost bifunction may not be monotone. We show that this problem can be solved by the inexact proximal point method if there exists a solution to the dual problem. An application of this approach to nonlinearly constrained problems is also suggested.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:119:y:2003:i:2:d:10.1023_b:jota.0000005448.12716.24
    DOI: 10.1023/B:JOTA.0000005448.12716.24
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    References listed on IDEAS

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    1. I. V. Konnov & S. Schaible, 2000. "Duality for Equilibrium Problems under Generalized Monotonicity," Journal of Optimization Theory and Applications, Springer, vol. 104(2), pages 395-408, February.
    2. Konnov, I.V. & Schaible, S., 1998. "Duality for Equilibrium Problems," The A. Gary Anderson Graduate School of Management 98-05, The A. Gary Anderson Graduate School of Management. University of California Riverside.
    3. N. El Farouq, 2001. "Pseudomonotone Variational Inequalities: Convergence of Proximal Methods," Journal of Optimization Theory and Applications, Springer, vol. 109(2), pages 311-326, May.
    4. N. El Farouq, 2001. "Pseudomonotone Variational Inequalities: Convergence of the Auxiliary Problem Method," Journal of Optimization Theory and Applications, Springer, vol. 111(2), pages 305-322, November.
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    Cited by:

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. J. Y. Bello Cruz & P. S. M. Santos & S. Scheimberg, 2013. "A Two-Phase Algorithm for a Variational Inequality Formulation of Equilibrium Problems," Journal of Optimization Theory and Applications, Springer, vol. 159(3), pages 562-575, December.
    6. 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.
    7. 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.
    8. Giancarlo Bigi & Mauro Passacantando, 2017. "Differentiated oligopolistic markets with concave cost functions via Ky Fan inequalities," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 40(1), pages 63-79, November.
    9. 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.
    10. 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.
    11. Nils Langenberg, 2012. "Interior point methods for equilibrium problems," Computational Optimization and Applications, Springer, vol. 53(2), pages 453-483, October.
    12. 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.
    13. 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.
    14. 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.
    15. 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.
    16. 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.
    17. 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.
    18. 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.
    19. 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.
    20. 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.
    21. Dang Hieu & Pham Kim Quy, 2023. "One-Step iterative method for bilevel equilibrium problem in Hilbert space," Journal of Global Optimization, Springer, vol. 85(2), pages 487-510, February.
    22. 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.
    23. 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.
    24. 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.

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