Journal of the Operations Research Society of China ›› 2021, Vol. 9 ›› Issue (3): 499-523.doi: 10.1007/s40305-020-00311-y

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Proximal Methods with Bregman Distances to Solve VIP on Hadamard Manifolds with Null Sectional Curvature

Erik Alex Papa Quiroz1, Paulo Roberto Oliveira2   

  1. 1 Universidad Nacional Mayor de San Marcos, Universidad Privada del Norte, Lima, Peru;
    2 PESC-COPPE, Federal University of Rio de Janeiro, Rio de Janeiro CEP 21941-972, Brazil
  • Received:2018-08-23 Revised:2020-05-05 Online:2021-09-30 Published:2021-09-26
  • Contact: Erik Alex Papa Quiroz,erikpapa@gmail.com;Paulo Roberto Oliveira,poliveir@cos.ufrj.br E-mail:erikpapa@gmail.com;poliveir@cos.ufrj.br
  • Supported by:
    This research was provided by INNOVATEPERU (Convenio No. 460-PNICP-BRI-2015).

Abstract: We present an extension of the proximal point method with Bregman distances to solve variational inequality problems (VIP) on Hadamard manifolds with null sectional curvature. Under some natural assumptions, as for example, the existence of solutions of the VIP and the monotonicity of the multivalued vector field, we prove that the sequence of the iterates given by the method converges to a solution of the problem. Furthermore, this convergence is linear or superlinear with respect to the Bregman distance.

Key words: Proximal point methods, Hadamard manifolds, Bregman distances, Variational inequality problems, Monotone vector field

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