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A refined proximal algorithm for nonconvex multiobjective optimization in Hilbert spaces

Author

Listed:
  • G. C. Bento

    (Federal University of Goiás)

  • J. X. Cruz Neto

    (Federal University of Piauí)

  • J. O. Lopes

    (Federal University of Piauí)

  • B. S. Mordukhovich

    (Wayne State University)

  • P. R. Silva Filho

    (Federal University of Piauí)

Abstract

This paper is devoted to general nonconvex problems of multiobjective optimization in Hilbert spaces. Based on limiting/Mordukhovich subgradients, we define a new notion of Pareto critical points for such problems, establish necessary optimality conditions for them, and then employ these conditions to develop a refined version of the vectorial proximal point algorithm providing its detailed convergence analysis. The obtained results largely extend those initiated by Bonnel et al. [SIAM J Optim, 15 (2005), pp. 953–970] for convex vector optimization problems, specifically in the case where the codomain is an m-dimensional space and by Bento et al. [SIAM J Optim, 28 (2018), pp. 1104-1120] for nonconvex finite-dimensional problems in terms of Clarke’s generalized gradients.

Suggested Citation

  • G. C. Bento & J. X. Cruz Neto & J. O. Lopes & B. S. Mordukhovich & P. R. Silva Filho, 2025. "A refined proximal algorithm for nonconvex multiobjective optimization in Hilbert spaces," Journal of Global Optimization, Springer, vol. 92(1), pages 187-203, May.
    • Handle: RePEc:spr:jglopt:v:92:y:2025:i:1:d:10.1007_s10898-024-01453-6
    DOI: 10.1007/s10898-024-01453-6
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    References listed on IDEAS

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    1. D. Aussel, 1998. "Subdifferential Properties of Quasiconvex and Pseudoconvex Functions: Unified Approach," Journal of Optimization Theory and Applications, Springer, vol. 97(1), pages 29-45, April.
    2. Glaydston Carvalho Bento & J.X. Cruz Neto & Antoine Soubeyran, 2014. "A Proximal Point-Type Method for Multicriteria Optimization," Post-Print hal-01463765, HAL.
    3. 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.
    4. 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.
    5. G. C. Bento & J. X. Cruz Neto & L. V. Meireles & A. Soubeyran, 2022. "Pareto solutions as limits of collective traps: an inexact multiobjective proximal point algorithm," Annals of Operations Research, Springer, vol. 316(2), pages 1425-1443, September.
    6. 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.
    7. 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.
    8. 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.
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