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Nonlinear conjugate gradient method for vector optimization on Riemannian manifolds with retraction and vector transport

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  • Chen, Kangming
  • Fukuda, Ellen Hidemi
  • Sato, Hiroyuki

Abstract

In this paper, we propose nonlinear conjugate gradient methods for vector optimization on Riemannian manifolds. The concepts of Wolfe and Zoutendjik conditions are extended to Riemannian manifolds. Specifically, the existence of intervals of step sizes that satisfy the Wolfe conditions is established. The convergence analysis covers the vector extensions of the Fletcher–Reeves, conjugate descent, and Dai–Yuan parameters. Under some assumptions, we prove that the sequence obtained by the proposed algorithm can converge to a Pareto stationary point. Moreover, several other choices of the parameter are discussed. Numerical experiments illustrating the practical behavior of the methods are presented.

Suggested Citation

  • Chen, Kangming & Fukuda, Ellen Hidemi & Sato, Hiroyuki, 2025. "Nonlinear conjugate gradient method for vector optimization on Riemannian manifolds with retraction and vector transport," Applied Mathematics and Computation, Elsevier, vol. 486(C).
    • Handle: RePEc:eee:apmaco:v:486:y:2025:i:c:s0096300324004624
    DOI: 10.1016/j.amc.2024.129001
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    References listed on IDEAS

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    1. Shahabeddin Najafi & Masoud Hajarian, 2023. "Multiobjective Conjugate Gradient Methods on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 197(3), pages 1229-1248, June.
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    5. Neculai Andrei, 2020. "Nonlinear Conjugate Gradient Methods for Unconstrained Optimization," Springer Optimization and Its Applications, Springer, number 978-3-030-42950-8, March.
    6. Ellen Fukuda & L. Graña Drummond, 2013. "Inexact projected gradient method for vector optimization," Computational Optimization and Applications, Springer, vol. 54(3), pages 473-493, April.
    7. Wang Chen & Xinmin Yang & Yong Zhao, 2023. "Conditional gradient method for vector optimization," Computational Optimization and Applications, Springer, vol. 85(3), pages 857-896, July.
    8. 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.
    9. Hiroyuki Sakai & Hideaki Iiduka, 2021. "Sufficient Descent Riemannian Conjugate Gradient Methods," Journal of Optimization Theory and Applications, Springer, vol. 190(1), pages 130-150, July.
    10. Neculai Andrei, 2020. "General Convergence Results for Nonlinear Conjugate Gradient Methods," Springer Optimization and Its Applications, in: Nonlinear Conjugate Gradient Methods for Unconstrained Optimization, chapter 0, pages 89-123, Springer.
    11. G. C. Bento & J. X. Cruz Neto & P. S. M. Santos, 2013. "An Inexact Steepest Descent Method for Multicriteria Optimization on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 159(1), pages 108-124, October.
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