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An extension of the proximal point algorithm with Bregman distances on Hadamard manifolds

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  • E. Papa Quiroz

Abstract

In this paper we present an extension of the proximal point algorithm with Bregman distances to solve constrained minimization problems with quasiconvex and convex objective function on Hadamard manifolds. The proposed algorithm is a modified and extended version of the one presented in Papa Quiroz and Oliveira (J Convex Anal 16(1): 49–69, 2009 ). An advantage of the proposed algorithm, for the nonconvex case, is that in each iteration the algorithm only needs to find a stationary point of the proximal function and not a global minimum. For that reason, from the computational point of view, the proposed algorithm is more practical than the earlier proximal method. Another advantage, for the convex case, is that using minimal condition on the problem data as well as on the proximal parameters we get the same convergence results of the Euclidean proximal algorithm using Bregman distances. Copyright Springer Science+Business Media New York 2013

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  • E. Papa Quiroz, 2013. "An extension of the proximal point algorithm with Bregman distances on Hadamard manifolds," Journal of Global Optimization, Springer, vol. 56(1), pages 43-59, May.
    • Handle: RePEc:spr:jglopt:v:56:y:2013:i:1:p:43-59
    DOI: 10.1007/s10898-012-9996-y
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    Cited by:

    1. 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.
    2. Alexandru Kristály & Chong Li & Genaro López-Acedo & Adriana Nicolae, 2016. "What Do ‘Convexities’ Imply on Hadamard Manifolds?," Journal of Optimization Theory and Applications, Springer, vol. 170(3), pages 1068-1074, September.
    3. G. C. Bento & J. X. Cruz Neto & P. A. Soares & A. Soubeyran, 2022. "A new regularization of equilibrium problems on Hadamard manifolds: applications to theories of desires," Annals of Operations Research, Springer, vol. 316(2), pages 1301-1318, September.

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