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General Versions of the Ekeland Variational Principle: Ekeland Points and Stop and Go Dynamics

Author

Listed:
  • Le Phuoc Hai

    (University of Science
    Vietnam National University)

  • Phan Quoc Khanh

    (Ton Duc Thang University)

  • Antoine Soubeyran

    (Aix-Marseille University)

Abstract

We establish general versions of the Ekeland variational principle (EVP), where we include two perturbation bifunctions to discuss and obtain better perturbations for obtaining three improved versions of the principle. Here, unlike the usual studies and applications of the EVP, which aim at exact minimizers via a limiting process, our versions provide good-enough approximate minimizers aiming at applications in particular situations. For the presentation of applications chosen in this paper, the underlying space is a partial quasi-metric one. To prove the aforementioned versions, we need a new proof technique. The novelties of the results are in both theoretical and application aspects. In particular, for applications, using our versions of the EVP together with new concepts of Ekeland points and stop and go dynamics, we study in detail human dynamics in terms of a psychological traveler problem, a typical model in behavioral sciences.

Suggested Citation

  • Le Phuoc Hai & Phan Quoc Khanh & Antoine Soubeyran, 2022. "General Versions of the Ekeland Variational Principle: Ekeland Points and Stop and Go Dynamics," Journal of Optimization Theory and Applications, Springer, vol. 195(1), pages 347-373, October.
    • Handle: RePEc:spr:joptap:v:195:y:2022:i:1:d:10.1007_s10957-022-02087-y
    DOI: 10.1007/s10957-022-02087-y
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    1. T. Q. Bao & S. Cobzaş & A. Soubeyran, 2018. "Variational principles, completeness and the existence of traps in behavioral sciences," Annals of Operations Research, Springer, vol. 269(1), pages 53-79, October.
    2. Phan Khanh & Dinh Quy, 2013. "Versions of Ekeland’s variational principle involving set perturbations," Journal of Global Optimization, Springer, vol. 57(3), pages 951-968, November.
    3. Majid Fakhar & Mohammadreza Khodakhah & Ali Mazyaki & Antoine Soubeyran & Jafar Zafarani, 2022. "Variational rationality, variational principles and the existence of traps in a changing environment," Journal of Global Optimization, Springer, vol. 82(1), pages 161-177, January.
    4. Guang-Ya Chen & X. Yang & Hui Yu, 2008. "Vector Ekeland’s variational principle in an F-type topological space," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 67(3), pages 471-478, June.
    5. Shahzad, Naseer & Valero, Oscar & Alghamdi, Mohammed A. & Alghamdi, Maryam A., 2015. "A fixed point theorem in partial quasi-metric spaces and an application to Software Engineering," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 1292-1301.
    6. Truong Q. Bao & Antoine Soubeyran, 2016. "Variational Analysis in Cone Pseudo-Quasimetric Spaces and Applications to Group Dynamics," Journal of Optimization Theory and Applications, Springer, vol. 170(2), pages 458-475, August.
    7. Truong Bao & Phan Khanh & Antoine Soubeyran, 2016. "Variational principles with generalized distances and the modelization of organizational change," Post-Print hal-01690191, HAL.
    8. T. Q. Bao & B. S. Mordukhovich & A. Soubeyran, 2015. "Variational Analysis in Psychological Modeling," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 290-315, January.
    9. Frank H. Clarke, 1976. "A New Approach to Lagrange Multipliers," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 165-174, May.
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    Cited by:

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    2. Antoine Soubeyran, 2022. "Variational rationality. Self regulation success as a succession of worthwhile moves that make sufficient progress," Working Papers hal-04041238, HAL.

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