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Proximal Point Method for Quasiconvex Functions in Riemannian Manifolds

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  • Erik Alex Papa Quiroz

    (Universidade Federal de Goiás
    Universidad Nacional Mayor de San Marcos
    Universidad Privada del Norte)

Abstract

This paper studies the convergence of the proximal point method for quasiconvex functions in finite dimensional complete Riemannian manifolds. We prove initially that, in the general case, when the objective function is proper and lower semicontinuous, each accumulation point of the sequence generated by the method, if it exists, is a limiting critical point of the function. Then, under the assumptions that the sectional curvature of the manifold is bounded above by some non negative constant and the objective function is quasiconvex we analyze two cases. When the constant is zero, the global convergence of the algorithm to a limiting critical point is assured and if it is positive, we prove the local convergence for a class of quasiconvex functions, which includes Lipschitz functions.

Suggested Citation

  • Erik Alex Papa Quiroz, 2024. "Proximal Point Method for Quasiconvex Functions in Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 202(3), pages 1268-1285, September.
    • Handle: RePEc:spr:joptap:v:202:y:2024:i:3:d:10.1007_s10957-024-02482-7
    DOI: 10.1007/s10957-024-02482-7
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    References listed on IDEAS

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    1. F. Lara, 2022. "On Strongly Quasiconvex Functions: Existence Results and Proximal Point Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 192(3), pages 891-911, March.
    2. David G. Luenberger, 1972. "The Gradient Projection Method Along Geodesics," Management Science, INFORMS, vol. 18(11), pages 620-631, July.
    3. Orizon Pereira Ferreira & Sándor Zoltán Németh & Lianghai Xiao, 2020. "On the Spherical Quasi-convexity of Quadratic Functions on Spherically Subdual Convex Sets," Journal of Optimization Theory and Applications, Springer, vol. 187(1), pages 1-21, October.
    4. 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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