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A new regularization of equilibrium problems on Hadamard manifolds: applications to theories of desires

Author

Listed:
  • Glaydston de Carvalho Bento

    (UFG - Universidade Federal de Goiás [Goiânia])

  • J.X. Cruz Neto

    (UFPI - Universidade Federal do Piauí)

  • A. Soares Jr Pedro

    (UFPI - Universidade Federal do Piauí)

  • Antoine Soubeyran

    (AMSE - Aix-Marseille Sciences Economiques - 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

In this paper, we introduce a new proximal algorithm for equilibrium problems on a genuine Hadamard manifold, using a new regularization term. We first extend recent existence results by considering pseudomonotone bifunctions and a weaker sufficient condition than the coercivity assumption. Then, we consider the convergence of this proximal-like algorithm which can be applied to genuinely Hadamard manifolds and not only to specific ones, as in the recent literature. A striking point is that our new regularization term have a clear interpretation in a recent "variational rationality" approach of human behavior. It represents the resistance to change aspects of such human dynamics driven by motivation to change aspects. This allows us to give an application to the theories of desires, showing how an agent must escape to a succession of temporary traps to be able to reach, at the end, his desires.

Suggested Citation

  • Glaydston de Carvalho Bento & J.X. Cruz Neto & A. Soares Jr Pedro & Antoine Soubeyran, 2022. "A new regularization of equilibrium problems on Hadamard manifolds: applications to theories of desires," Post-Print hal-03530791, HAL.
    • Handle: RePEc:hal:journl:hal-03530791
    DOI: 10.1007/s10479-021-04052-w
    Note: View the original document on HAL open archive server: https://amu.hal.science/hal-03530791
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    References listed on IDEAS

    as
    1. J. X. Cruz Neto & P. R. Oliveira & P. A. Soares & A. Soubeyran, 2014. "Proximal Point Method on Finslerian Manifolds and the “Effort–Accuracy” Trade-off," Journal of Optimization Theory and Applications, Springer, vol. 162(3), pages 873-891, September.
    2. 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.
    3. Vladimir Bulavsky & Vyacheslav Kalashnikov, 1998. "A Newton-like approach to solvingan equilibrium problem," Annals of Operations Research, Springer, vol. 81(0), pages 115-128, June.
    4. Gayatri Pany & Ram N. Mohapatra & Sabyasachi Pani, 2018. "Solution of a class of equilibrium problems and variational inequalities in FC spaces," Annals of Operations Research, Springer, vol. 269(1), pages 565-582, October.
    5. Francisco Facchinei & Christian Kanzow, 2010. "Generalized Nash Equilibrium Problems," Annals of Operations Research, Springer, vol. 175(1), pages 177-211, March.
    6. Glaydston C. Bento & Jefferson G. Melo, 2012. "Subgradient Method for Convex Feasibility on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 773-785, March.
    7. O. Ferreira & L. Pérez & S. Németh, 2005. "Singularities of Monotone Vector Fields and an Extragradient-type Algorithm," Journal of Global Optimization, Springer, vol. 31(1), pages 133-151, January.
    8. Xiaomin Zhang & Zezhong Wu, 2013. "Optimality Conditions and Duality of Three Kinds of Nonlinear Fractional Programming Problems," Advances in Operations Research, Hindawi, vol. 2013, pages 1-9, November.
    9. Edvaldo E. A. Batista & Glaydston de Carvalho Bento & Orizon P. Ferreira, 2016. "Enlargement of Monotone Vector Fields and an Inexact Proximal Point Method for Variational Inequalities in Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 916-931, September.
    10. Jianke Zhang, 2013. "Optimality Condition and Wolfe Duality for Invex Interval-Valued Nonlinear Programming Problems," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-11, December.
    11. E. Papa Quiroz, 2013. "An extension of the proximal point algorithm with Bregman distances on Hadamard manifolds," Journal of Global Optimization, Springer, vol. 56(1), pages 43-59, May.
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    Cited by:

    1. Glaydston C. Bento & João X. Cruz Neto & Jurandir O. Lopes & Ítalo D. L. Melo & Pedro Silva Filho, 2024. "A New Approach About Equilibrium Problems via Busemann Functions," Journal of Optimization Theory and Applications, Springer, vol. 200(1), pages 428-436, January.

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