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Pareto solutions as limits of collective traps: an inexact multiobjective proximal point algorithm

Author

Listed:
  • Glaydston Carvalho Bento

    (UFG - Universidade Federal de Goiás [Goiânia])

  • João Xavier da Cruz Neto

    (UFPI - Universidade Federal do Piauí)

  • L. Meireles

    (IFGOIANO - Instituto Federal Goiano = Goiano Federal Institute)

  • 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 definition of approximate Pareto efficient solution as well as a necessary condition for such solutions in the multiobjective setting on Riemannian manifolds. We also propose an inexact proximal point method for nonsmooth multiobjective optimization in the Riemannian context by using the notion of approximate solution. The main convergence result ensures that each cluster point (if any) of any sequence generated by the method is a Pareto critical point. Furthermore, when the problem is convex on a Hadamard manifold, full convergence of the method for a weak Pareto efficient solution is obtained. As an application, we show how a Pareto critical point can be reached as a limit of traps in the context of the variational rationality approach of stay and change human dynamics.

Suggested Citation

  • Glaydston Carvalho Bento & João Xavier da Cruz Neto & L. Meireles & Antoine Soubeyran, 2022. "Pareto solutions as limits of collective traps: an inexact multiobjective proximal point algorithm," Post-Print hal-03680291, HAL.
    • Handle: RePEc:hal:journl:hal-03680291
    DOI: 10.1007/s10479-022-04719-y
    Note: View the original document on HAL open archive server: https://amu.hal.science/hal-03680291
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    References listed on IDEAS

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    1. Ceng, Lu-Chuan & Yao, Jen-Chih, 2007. "Approximate proximal methods in vector optimization," European Journal of Operational Research, Elsevier, vol. 183(1), pages 1-19, November.
    2. G. C. Bento & J. X. Cruz Neto, 2013. "A Subgradient Method for Multiobjective Optimization on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 125-137, October.
    3. G. Bento & J. Cruz Neto & G. López & Antoine Soubeyran & J. Souza, 2018. "The Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization with Application to the Compromise Problem," Post-Print hal-01985333, HAL.
    4. L. C. Ceng & B. S. Mordukhovich & J. C. Yao, 2010. "Hybrid Approximate Proximal Method with Auxiliary Variational Inequality for Vector Optimization," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 267-303, August.
    5. Glaydston de C. Bento & João Xavier Cruz Neto & Lucas V. Meireles, 2018. "Proximal Point Method for Locally Lipschitz Functions in Multiobjective Optimization of Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 179(1), pages 37-52, October.
    6. X. X. Huang & X. Q. Yang, 2004. "Duality for Multiobjective Optimization via Nonlinear Lagrangian Functions," Journal of Optimization Theory and Applications, Springer, vol. 120(1), pages 111-127, January.
    7. 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.
    8. Ronaldo Gregório & Paulo Oliveira, 2011. "A logarithmic-quadratic proximal point scalarization method for multiobjective programming," Journal of Global Optimization, Springer, vol. 49(2), pages 281-291, February.
    9. 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.
    10. Villacorta, Kely D.V. & Oliveira, P. Roberto, 2011. "An interior proximal method in vector optimization," European Journal of Operational Research, Elsevier, vol. 214(3), pages 485-492, November.
    11. G. C. Bento & O. P. Ferreira & P. R. Oliveira, 2012. "Unconstrained Steepest Descent Method for Multicriteria Optimization on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 88-107, July.
    12. Erik Alex Papa Quiroz & Nancy Baygorrea Cusihuallpa & Nelson Maculan, 2020. "Inexact Proximal Point Methods for Multiobjective Quasiconvex Minimization on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 186(3), pages 879-898, September.
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