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Subgradient Projection Algorithms for Convex Feasibility on Riemannian Manifolds with Lower Bounded Curvatures

Author

Listed:
  • X. M. Wang

    (Zhejiang University
    Guizhou University)

  • C. Li

    (Zhejiang University)

  • J. C. Yao

    (Kaohsiung Medical University
    King Abdulaziz University)

Abstract

Under the assumption that the sectional curvature of the manifold is bounded from below, we establish convergence result about the cyclic subgradient projection algorithm for convex feasibility problem presented in a paper by Bento and Melo on Riemannian manifolds (J Optim Theory Appl 152, 773–785, 2012). If, additionally, we assume that a Slater type condition is satisfied, then we further show that, without changing the step size, this algorithm terminates in a finite number of iterations. Clearly, our results extend the corresponding ones due to Bento and Melo and, in particular, we solve partially the open problem proposed in the paper by Bento and Melo.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:164:y:2015:i:1:d:10.1007_s10957-014-0568-9
    DOI: 10.1007/s10957-014-0568-9
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    References listed on IDEAS

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    1. 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.
    2. David G. Luenberger, 1972. "The Gradient Projection Method Along Geodesics," Management Science, INFORMS, vol. 18(11), pages 620-631, July.
    3. 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.
    4. 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.
    5. 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.
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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. 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.
    3. Yldenilson Torres Almeida & João Xavier Cruz Neto & Paulo Roberto Oliveira & João Carlos de Oliveira Souza, 2020. "A modified proximal point method for DC functions on Hadamard manifolds," Computational Optimization and Applications, Springer, vol. 76(3), pages 649-673, July.
    4. 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.
    5. Ítalo Dowell Lira Melo & João Xavier Cruz Neto & José Márcio Machado Brito, 2022. "Strong Convergence of Alternating Projections," Journal of Optimization Theory and Applications, Springer, vol. 194(1), pages 306-324, July.
    6. 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.
    7. 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.

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