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Local proximal algorithms in Riemannian manifolds: Application to the behavioral traveler's problem

Author

Listed:
  • Erik Papa Quiroz

    (UFG - Universidade Federal de Goiás [Goiânia], PUCP - Pontificia Universidad Católica del Perú = Pontifical Catholic University of Peru)

  • 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

Local proximal point algorithms with quasi distances to find critical points (or minimizer points in the convex case) of functions in finite dimensional Riemannian manifolds are introduced. We prove that bounded sequences of the algorithm generated by proper bounded from below, lower semicontinuous and locally Lipschitz functions have accumulation points which are critical points (minimizer points in the convex case). Moreover, for KurdykaLojasiewicz functions, the sequence globally converges to a critical point. We applied the algorithm to a behavioral traveler's problem where an individual tries to satisfy locally his needs and desires by moving from one city to the next, with costs to move playing a major role.

Suggested Citation

  • Erik Papa Quiroz & Antoine Soubeyran, 2024. "Local proximal algorithms in Riemannian manifolds: Application to the behavioral traveler's problem," Post-Print hal-04930974, HAL.
    • Handle: RePEc:hal:journl:hal-04930974
    DOI: 10.3934/eect.2024072
    Note: View the original document on HAL open archive server: https://hal.science/hal-04930974v1
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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. Eric J. Johnson & John W. Payne, 1985. "Effort and Accuracy in Choice," Management Science, INFORMS, vol. 31(4), pages 395-414, April.
    3. Majid Fakhar & Mohammadreza Khodakhah & Antoine Soubeyran & Jafar Zafarani, 2022. "Set-valued variational principles. When migration improves quality of life," Post-Print hal-03659698, HAL.
    4. 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.
    5. M. Farrokhiniya & A. Barani, 2020. "Limiting Subdifferential Calculus and Perturbed Distance Function in Riemannian Manifolds," Journal of Global Optimization, Springer, vol. 77(3), pages 661-685, July.
    6. Truong Quang Bao & Antoine Soubeyran, 2019. "Variational principles in set optimization with domination structures and application to changing jobs," Post-Print hal-02497051, HAL.
    7. 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.
    8. G. C. Bento & A. Soubeyran, 2015. "Generalized Inexact Proximal Algorithms: Routine’s Formation with Resistance to Change, Following Worthwhile Changes," Journal of Optimization Theory and Applications, Springer, vol. 166(1), pages 172-187, July.
    9. E. A. Papa Quiroz & A. Soubeyran & P. R. Oliveira, 2023. "Coercivity and generalized proximal algorithms: application—traveling around the world," Annals of Operations Research, Springer, vol. 321(1), pages 451-467, February.
    10. Erik Alex Papa Quiroz & Antoine Soubeyran & Paulo Roberto Oliveira, 2023. "Coercivity and generalized proximal algorithms: application—traveling around the world," Post-Print hal-03665851, HAL.
    11. Glaydston Carvalho Bento & João Xavier Cruz Neto & Paulo Roberto Oliveira, 2016. "A New Approach to the Proximal Point Method: Convergence on General Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 743-755, March.
    12. H. Apolinário & E. Papa Quiroz & P. Oliveira, 2016. "A scalarization proximal point method for quasiconvex multiobjective minimization," Journal of Global Optimization, Springer, vol. 64(1), pages 79-96, January.
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