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Iteration-Complexity of Gradient, Subgradient and Proximal Point Methods on Riemannian Manifolds

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  • Glaydston C. Bento

    (IME, Universidade Federal de Goiás)

  • Orizon P. Ferreira

    (IME, Universidade Federal de Goiás)

  • Jefferson G. Melo

    (IME, Universidade Federal de Goiás)

Abstract

This paper considers optimization problems on Riemannian manifolds and analyzes the iteration-complexity for gradient and subgradient methods on manifolds with nonnegative curvatures. By using tools from Riemannian convex analysis and directly exploring the tangent space of the manifold, we obtain different iteration-complexity bounds for the aforementioned methods, thereby complementing and improving related results. Moreover, we also establish an iteration-complexity bound for the proximal point method on Hadamard manifolds.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:173:y:2017:i:2:d:10.1007_s10957-017-1093-4
    DOI: 10.1007/s10957-017-1093-4
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    References listed on IDEAS

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    1. 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.
    2. 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.
    3. 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.
    4. David G. Luenberger, 1972. "The Gradient Projection Method Along Geodesics," Management Science, INFORMS, vol. 18(11), pages 620-631, July.
    5. 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.
    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. NESTEROV , Yu. & TODD, Mike, 2002. "On the Riemannian geometry defined by self-concordant barriers and interior-point methods," LIDAM Reprints CORE 1595, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    8. 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.
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    Cited by:

    1. 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.
    2. Geovani N. Grapiglia & Gabriel F. D. Stella, 2023. "An Adaptive Riemannian Gradient Method Without Function Evaluations," Journal of Optimization Theory and Applications, Springer, vol. 197(3), pages 1140-1160, June.
    3. Harry Oviedo, 2023. "Proximal Point Algorithm with Euclidean Distance on the Stiefel Manifold," Mathematics, MDPI, vol. 11(11), pages 1-17, May.
    4. 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.
    5. 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.

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