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

  • Glaydston de Carvalho Bento
  • João Xavier da Cruz Neto
  • Paulo Roberto Oliveira


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.


Proximal method Non-convex optimization Kurdyka–Lojasiewicz inequality Riemannian manifolds 

Mathematics Subject Classification

49J52 65K05 58C99 90C26 90C56 



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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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Glaydston de Carvalho Bento
    • 1
  • João Xavier da Cruz Neto
    • 2
  • Paulo Roberto Oliveira
    • 3
  1. 1.IMEUniversidade Federal de GoiásGoiâniaBrazil
  2. 2.DMUniversidade Federal do PiauíTerezinaBrazil
  3. 3.COPPE-SistemasUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil

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