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A New Approach to the Proximal Point Method: Convergence on General Riemannian Manifolds

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Abstract

In this paper, we present a new approach to the proximal point method in the Riemannian context. In particular, without requiring any restrictive assumptions about the sign of the sectional curvature of the manifold, we obtain full convergence for any bounded sequence generated by the proximal point method, in the case that the objective function satisfies the Kurdyka–Lojasiewicz inequality. In our approach, we extend the applicability of the proximal point method to be able to solve any problem that can be formulated as the minimizing of a definable function, such as one that is analytic, restricted to a compact manifold, on which the sign of the sectional curvature is not necessarily constant.

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Acknowledgments

G. C. Bento was supported in part by CAPES-MES-CUBA 226/2012, FAPEG 201210267000909-05/2012 and CNPq Grants 458479/2014-4, 471815/2012-8, 303732/2011-3, 312077/ 2014-9., J. X. Cruz Neto was partially supported by CNPq GRANT 305462/2014-8, and P. R. Oliveira was supported in part by CNPq.

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Authors and Affiliations

  1. IME, Universidade Federal de Goiás, Goiânia, GO, 74001-970, Brazil

    Glaydston de Carvalho Bento

  2. DM, Universidade Federal do Piauí, Terezina, PI, 64049-550, Brazil

    João Xavier da Cruz Neto

  3. COPPE-Sistemas, Universidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, 21945-970, Brazil

    Paulo Roberto Oliveira

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  1. Glaydston de Carvalho Bento
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  2. João Xavier da Cruz Neto
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  3. Paulo Roberto Oliveira
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Correspondence to Glaydston de Carvalho Bento.

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Communicated by Sándor Zoltán Németh.

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de Carvalho Bento, G., da Cruz Neto, J.X. & Oliveira, P.R. A New Approach to the Proximal Point Method: Convergence on General Riemannian Manifolds. J Optim Theory Appl 168, 743–755 (2016). https://doi.org/10.1007/s10957-015-0861-2

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