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Proximal Point Method for Quasiconvex Functions in Riemannian Manifolds

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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.

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Acknowledgements

I am grateful to the Optimization Research Group at Institute of Mathematics and Statistic of the Federal Goias University (IME-UFG), Brazil, by the suggestions developed during the Optimization Seminars. In particular to Orizon Ferreira and Glaydston Bento who give me some observations on the paper.

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Authors and Affiliations

  1. Universidade Federal de Goiás, Goiânia, Brazil

    Erik Alex Papa Quiroz

  2. Universidad Nacional Mayor de San Marcos, Lima, Peru

    Erik Alex Papa Quiroz

  3. Universidad Privada del Norte, Lima, Peru

    Erik Alex Papa Quiroz

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  1. Erik Alex Papa Quiroz
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Correspondence to Erik Alex Papa Quiroz.

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Communicated by Nicolas Hadjisavvas.

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Quiroz, E.A.P. Proximal Point Method for Quasiconvex Functions in Riemannian Manifolds. J Optim Theory Appl 202, 1268–1285 (2024). https://doi.org/10.1007/s10957-024-02482-7

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  • DOI: https://doi.org/10.1007/s10957-024-02482-7

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