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Subgradient Algorithm on Riemannian Manifolds

Author

Listed:
  • O. P. Ferreira

    (Universidade Federal de Goiás)

  • P. R. Oliveira

    (Universidade Federal do Rio de Janeiro)

Abstract

The subgradient method is generalized to the context of Riemannian manifolds. The motivation can be seen in non-Euclidean metrics that occur in interior-point methods. In that frame, the natural curves for local steps are the geodesies relative to the specific Riemannian manifold. In this paper, the influence of the sectional curvature of the manifold on the convergence of the method is discussed, as well as the proof of convergence if the sectional curvature is nonnegative.

Suggested Citation

  • O. P. Ferreira & P. R. Oliveira, 1998. "Subgradient Algorithm on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 97(1), pages 93-104, April.
    • Handle: RePEc:spr:joptap:v:97:y:1998:i:1:d:10.1023_a:1022675100677
    DOI: 10.1023/A:1022675100677
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    Citations

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    Cited by:

    1. Glaydston C. Bento & Orizon P. Ferreira & Jefferson G. Melo, 2017. "Iteration-Complexity of Gradient, Subgradient and Proximal Point Methods on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 548-562, May.
    2. E. A. Papa Quiroz & P. R. Oliveira, 2007. "New Self-Concordant Barrier for the Hypercube," Journal of Optimization Theory and Applications, Springer, vol. 135(3), pages 475-490, December.
    3. Qinsi Wang & Wei Hong Yang, 2024. "An adaptive regularized proximal Newton-type methods for composite optimization over the Stiefel manifold," Computational Optimization and Applications, Springer, vol. 89(2), pages 419-457, November.
    4. Dewei Zhang & Sam Davanloo Tajbakhsh, 2023. "Riemannian Stochastic Variance-Reduced Cubic Regularized Newton Method for Submanifold Optimization," Journal of Optimization Theory and Applications, Springer, vol. 196(1), pages 324-361, January.
    5. 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.
    6. João Carlos de O. Souza, 2018. "Proximal Point Methods for Lipschitz Functions on Hadamard Manifolds: Scalar and Vectorial Cases," Journal of Optimization Theory and Applications, Springer, vol. 179(3), pages 745-760, December.
    7. X. M. Wang & C. Li & J. C. Yao, 2015. "Subgradient Projection Algorithms for Convex Feasibility on Riemannian Manifolds with Lower Bounded Curvatures," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 202-217, January.
    8. Peng Zhang & Gejun Bao, 2018. "An Incremental Subgradient Method on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 711-727, March.
    9. da Silva Alves, Charlan Dellon & Oliveira, Paulo Roberto & Gregório, Ronaldo Malheiros, 2021. "Lα Riemannian weighted centers of mass applied to compose an image filter to diffusion tensor imaging," Applied Mathematics and Computation, Elsevier, vol. 390(C).
    10. O. Ferreira & A. Iusem & S. Németh, 2014. "Concepts and techniques of optimization on the sphere," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 22(3), pages 1148-1170, October.
    11. Lei Wang & Xin Liu & Yin Zhang, 2023. "A communication-efficient and privacy-aware distributed algorithm for sparse PCA," Computational Optimization and Applications, Springer, vol. 85(3), pages 1033-1072, July.
    12. Glaydston C. Bento & Jefferson G. Melo, 2012. "Subgradient Method for Convex Feasibility on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 152(3), pages 773-785, March.
    13. Xiao-bo Li & Li-wen Zhou & Nan-jing Huang, 2016. "Gap Functions and Global Error Bounds for Generalized Mixed Variational Inequalities on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 830-849, March.
    14. Guo-ji Tang & Nan-jing Huang, 2012. "Korpelevich’s method for variational inequality problems on Hadamard manifolds," Journal of Global Optimization, Springer, vol. 54(3), pages 493-509, November.
    15. Gregório, R.M. & Oliveira, P.R. & Alves, C.D.S., 2019. "A two-phase-like proximal point algorithm in domains of positivity," Applied Mathematics and Computation, Elsevier, vol. 343(C), pages 67-89.
    16. J. Souza & P. Oliveira, 2015. "A proximal point algorithm for DC fuctions on Hadamard manifolds," Journal of Global Optimization, Springer, vol. 63(4), pages 797-810, December.
    17. J. X. Cruz Neto & F. M. O. Jacinto & P. A. Soares & J. C. O. Souza, 2018. "On maximal monotonicity of bifunctions on Hadamard manifolds," Journal of Global Optimization, Springer, vol. 72(3), pages 591-601, November.
    18. 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.
    19. João Xavier da Cruz Neto & Ítalo Dowell Lira Melo & Paulo Alexandre Araújo Sousa, 2017. "Convexity and Some Geometric Properties," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 459-470, May.

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