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On the Relationship Between the Kurdyka–Łojasiewicz Property and Error Bounds on Hadamard Manifolds

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Listed:
  • João Xavier Cruz Neto

    (Federal University of Piauí)

  • Ítalo Dowell Lira Melo

    (Federal University of Piauí)

  • Paulo Alexandre Sousa

    (Federal University of Piauí)

  • João Carlos Oliveira Souza

    (Federal University of Piauí)

Abstract

This paper studies the interplay between the concepts of error bounds and the Kurdyka–Łojasiewicz (KL) inequality on Hadamard manifolds. To this end, we extend some properties and existence results of a solution for differential inclusions on Hadamard manifolds. As a second contribution, we show how the KL inequality can be used to obtain the convergence of the gradient method for solving convex feasibility problems on Hadamard manifolds. The convergence results of the alternating projection method are also established for cyclic and random projections on Hadamard manifolds and, more generally, CAT(0) spaces.

Suggested Citation

  • João Xavier Cruz Neto & Ítalo Dowell Lira Melo & Paulo Alexandre Sousa & João Carlos Oliveira Souza, 2024. "On the Relationship Between the Kurdyka–Łojasiewicz Property and Error Bounds on Hadamard Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 200(3), pages 1255-1285, March.
    • Handle: RePEc:spr:joptap:v:200:y:2024:i:3:d:10.1007_s10957-024-02386-6
    DOI: 10.1007/s10957-024-02386-6
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    References listed on IDEAS

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    1. David G. Luenberger, 1972. "The Gradient Projection Method Along Geodesics," Management Science, INFORMS, vol. 18(11), pages 620-631, July.
    2. 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.
    3. Xiangmei Wang & Chong Li & Jen-Chih Yao, 2016. "On Some Basic Results Related to Affine Functions on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 783-803, September.
    4. Joao Xavier Cruz Neto & Paulo Roberto Oliveira & A. Soares Jr Pedro & Antoine Soubeyran, 2013. "Learning how to Play Nash, Potential Games and Alternating Minimization Method for Structured Nonconvex Problems on Riemannian Manifolds," Post-Print hal-01500875, HAL.
    5. 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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