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Dual Descent Methods as Tension Reduction Systems

Author

Listed:
  • Glaydston de Carvalho Bento

    (UFG - Universidade Federal de Goiás [Goiânia])

  • João Xavier da Cruz Neto

    (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)

  • Valdinês Leite de Sousa Júnior

    (UFG - Universidade Federal de Goiás [Goiânia])

Abstract

In this paper, driven by applications in Behavioral Sciences, wherein the speed of convergence matters considerably, we compare the speed of convergence of two descent methods for functions that satisfy the well-known Kurdyka–Lojasiewicz property in a quasi-metric space. This includes the extensions to a quasi-metric space of both the primal and dual descent methods. While the primal descent method requires the current step to be more or less half of the size of the previous step, the dual approach considers more or less half of the previous decrease in the objective function to be minimized. We provide applications to the famous "Tension systems approach" in Psychology.

Suggested Citation

  • Glaydston de Carvalho Bento & João Xavier da Cruz Neto & Antoine Soubeyran & Valdinês Leite de Sousa Júnior, 2016. "Dual Descent Methods as Tension Reduction Systems," Post-Print hal-01690176, HAL.
    • Handle: RePEc:hal:journl:hal-01690176
    DOI: 10.1007/s10957-016-0994-y
    Note: View the original document on HAL open archive server: https://amu.hal.science/hal-01690176
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    References listed on IDEAS

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    1. Hédy Attouch & Jérôme Bolte & Patrick Redont & Antoine Soubeyran, 2010. "Proximal Alternating Minimization and Projection Methods for Nonconvex Problems: An Approach Based on the Kurdyka-Łojasiewicz Inequality," Mathematics of Operations Research, INFORMS, vol. 35(2), pages 438-457, May.
    2. J. Y. Bello Cruz & G. Bouza Allende, 2014. "A Steepest Descent-Like Method for Variable Order Vector Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 162(2), pages 371-391, August.
    3. 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.
    4. Marc Fuentes & Jérôme Malick & Claude Lemaréchal, 2012. "Descentwise inexact proximal algorithms for smooth optimization," Computational Optimization and Applications, Springer, vol. 53(3), pages 755-769, December.
    5. Truong Q. Bao & Boris S. Mordukhovich & Antoine Soubeyran, 2015. "Minimal points, variational principles, and variable preferences in set optimization," Post-Print hal-01457319, HAL.
    6. 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.
    7. 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.
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    Cited by:

    1. Glaydston de Carvalho Bento & Orizon Pereira Ferreira & Antoine Soubeyran & Valdinês Leite de Sousa Júnior, 2018. "Inexact Multi-Objective Local Search Proximal Algorithms: Application to Group Dynamic and Distributive Justice Problems," Journal of Optimization Theory and Applications, Springer, vol. 177(1), pages 181-200, April.

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