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Strong Convergence of Alternating Projections

Author

Listed:
  • Ítalo Dowell Lira Melo

    (Universidade Federal do Piauí)

  • João Xavier Cruz Neto

    (Universidade Federal do Piauí)

  • José Márcio Machado Brito

    (Universidade Federal do Piauí)

Abstract

In this paper, we provide a necessary and sufficient condition under which the method of alternating projections on Hadamard spaces converges strongly. This result is new even in the context of Hilbert spaces. In particular, we found the circumstance under which the iteration of a point by projections converges strongly and we answer partially the main question that motivated Bruck’s paper (J Math Anal Appl 88:319–322, 1982). We apply this condition to generalize Prager’s theorem for Hadamard manifolds and generalize Sakai’s theorem for a larger class of the sequences with full measure with respect to Bernoulli measure. In particular, we answer to a long-standing open problem concerning the convergence of the successive projection method (Aleyner and Reich in J Convex Anal 16:633–640, 2009). Furthermore, we study the method of alternating projections for a nested decreasing sequence of convex sets on Hadamard manifolds, and we obtain an alternative proof of the convergence of the proximal point method.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:194:y:2022:i:1:d:10.1007_s10957-022-02028-9
    DOI: 10.1007/s10957-022-02028-9
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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. David Ariza-Ruiz & Genaro López-Acedo & Adriana Nicolae, 2015. "The Asymptotic Behavior of the Composition of Firmly Nonexpansive Mappings," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 409-429, November.
    3. 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.
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    Cited by:

    1. Daylen K. Thimm, 2024. "Most Iterations of Projections Converge," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 285-304, October.

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