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Subgradient Method for Convex Feasibility on Riemannian Manifolds

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  • Glaydston C. Bento

    (IME-Universidade Federal de Goiás)

  • Jefferson G. Melo

    (IME-Universidade Federal de Goiás)

Abstract

In this paper, a subgradient type algorithm for solving convex feasibility problem on Riemannian manifold is proposed and analysed. The sequence generated by the algorithm converges to a solution of the problem, provided the sectional curvature of the manifold is non-negative. Moreover, assuming a Slater type qualification condition, we analyse a variant of the first algorithm, which generates a sequence with finite convergence property, i.e., a feasible point is obtained after a finite number of iterations. Some examples motivating the application of the algorithm for feasibility problems, nonconvex in the usual sense, are considered.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:152:y:2012:i:3:d:10.1007_s10957-011-9921-4
    DOI: 10.1007/s10957-011-9921-4
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. Regina Burachik & Alfredo Iusem & Jefferson Melo, 2010. "A primal dual modified subgradient algorithm with sharp Lagrangian," Journal of Global Optimization, Springer, vol. 46(3), pages 347-361, March.
    4. J. H. Wang, 2011. "Convergence of Newton’s Method for Sections on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 148(1), pages 125-145, January.
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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. Glaydston Carvalho Bento & João Xavier Cruz Neto & Paulo Roberto Oliveira, 2016. "A New Approach to the Proximal Point Method: Convergence on General Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 743-755, March.
    3. 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.
    4. 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.
    5. Savin Treanţă & Balendu Bhooshan Upadhyay & Arnav Ghosh & Kamsing Nonlaopon, 2022. "Optimality Conditions for Multiobjective Mathematical Programming Problems with Equilibrium Constraints on Hadamard Manifolds," Mathematics, MDPI, vol. 10(19), pages 1-20, September.
    6. 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.
    7. Henri Bonnel & Léonard Todjihoundé & Constantin Udrişte, 2015. "Semivectorial Bilevel Optimization on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 464-486, November.
    8. 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.
    9. Balendu Bhooshan Upadhyay & Arnav Ghosh, 2023. "On Constraint Qualifications for Mathematical Programming Problems with Vanishing Constraints on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 199(1), pages 1-35, October.
    10. 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.
    11. 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.
    12. G. C. Bento & J. X. Cruz Neto & P. A. Soares & A. Soubeyran, 2022. "A new regularization of equilibrium problems on Hadamard manifolds: applications to theories of desires," Annals of Operations Research, Springer, vol. 316(2), pages 1301-1318, September.
    13. Edvaldo E. A. Batista & Glaydston de Carvalho Bento & Orizon P. Ferreira, 2016. "Enlargement of Monotone Vector Fields and an Inexact Proximal Point Method for Variational Inequalities in Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 916-931, September.
    14. M. Farrokhiniya & A. Barani, 2020. "Limiting Subdifferential Calculus and Perturbed Distance Function in Riemannian Manifolds," Journal of Global Optimization, Springer, vol. 77(3), pages 661-685, July.
    15. 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.

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