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Convergence of Inexact Steepest Descent Algorithm for Multiobjective Optimizations on Riemannian Manifolds Without Curvature Constraints

Author

Listed:
  • X. M. Wang

    (Guizhou University)

  • J. H. Wang

    (Hangzhou Normal University)

  • C. Li

    (Zhejiang University)

Abstract

We study the issue of convergence for inexact steepest descent algorithm (employing general step sizes) for multiobjective optimizations on general Riemannian manifolds (without curvature constraints). Under the assumption of the local convexity/quasi-convexity, local/global convergence results are established. Furthermore, without the assumption of the local convexity/quasi-convexity, but under an error bound-like condition, local/global convergence results and convergence rate estimates are presented, which are new even in the linear space setting. Our results improve/extend the corresponding ones in (Wang et al. in SIAM J Optim 31(1):172–199, 2021) for scalar optimization problems on Riemannian manifolds to multiobjective ones. Finally, for the special case when the inexact steepest descent algorithm employing Armijo rule, our results improve/extend the corresponding ones in (Ferreira et al. in J Optim Theory Appl 184:507–533, 2020) by relaxing curvature constraints.

Suggested Citation

  • X. M. Wang & J. H. Wang & C. Li, 2023. "Convergence of Inexact Steepest Descent Algorithm for Multiobjective Optimizations on Riemannian Manifolds Without Curvature Constraints," Journal of Optimization Theory and Applications, Springer, vol. 198(1), pages 187-214, July.
    • Handle: RePEc:spr:joptap:v:198:y:2023:i:1:d:10.1007_s10957-023-02235-y
    DOI: 10.1007/s10957-023-02235-y
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    References listed on IDEAS

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    1. Orizon P. Ferreira & Mauricio S. Louzeiro & Leandro F. Prudente, 2020. "Iteration-Complexity and Asymptotic Analysis of Steepest Descent Method for Multiobjective Optimization on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 184(2), pages 507-533, February.
    2. 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.
    3. Y. Yang, 2007. "Globally Convergent Optimization Algorithms on Riemannian Manifolds: Uniform Framework for Unconstrained and Constrained Optimization," Journal of Optimization Theory and Applications, Springer, vol. 132(2), pages 245-265, February.
    4. J. H. Wang & G. López & V. Martín-Márquez & C. Li, 2010. "Monotone and Accretive Vector Fields on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 146(3), pages 691-708, September.
    5. 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.
    6. Jörg Fliege & Benar Fux Svaiter, 2000. "Steepest descent methods for multicriteria optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 51(3), pages 479-494, August.
    7. O. Ferreira & L. Pérez & S. Németh, 2005. "Singularities of Monotone Vector Fields and an Extragradient-type Algorithm," Journal of Global Optimization, Springer, vol. 31(1), pages 133-151, January.
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