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

Erik Alex Papa Quiroz; Paulo Roberto Oliveira

doi:10.1007/s40305-020-00311-y

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

Erik Alex Papa Quiroz
, Paulo Roberto Oliveira

Universidad Nacional Mayor de San Marcos
Universidade Federal do Rio de Janeiro

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	499-523
Number of pages	25
Journal	Journal of the Operations Research Society of China
Volume	9
Issue number	3
DOIs	https://doi.org/10.1007/s40305-020-00311-y
State	Published - Sep 2021
Externally published	Yes

Keywords

Bregman distances
Hadamard manifolds
Monotone vector field
Proximal point methods
Variational inequality problems

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10.1007/s40305-020-00311-y

Cite this

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title = "Proximal Methods with Bregman Distances to Solve VIP on Hadamard Manifolds with Null Sectional Curvature",

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.",

keywords = "Bregman distances, Hadamard manifolds, Monotone vector field, Proximal point methods, Variational inequality problems",

author = "\{Papa Quiroz\}, \{Erik Alex\} and Oliveira, \{Paulo Roberto\}",

note = "Publisher Copyright: {\textcopyright} 2020, Operations Research Society of China, Periodicals Agency of Shanghai University, Science Press, and Springer-Verlag GmbH Germany, part of Springer Nature.",

year = "2021",

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T1 - Proximal Methods with Bregman Distances to Solve VIP on Hadamard Manifolds with Null Sectional Curvature

AU - Papa Quiroz, Erik Alex

AU - Oliveira, Paulo Roberto

PY - 2021/9

Y1 - 2021/9

N2 - 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.

AB - 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.

KW - Bregman distances

KW - Hadamard manifolds

KW - Monotone vector field

KW - Proximal point methods

KW - Variational inequality problems

UR - https://www.scopus.com/pages/publications/85089312376

U2 - 10.1007/s40305-020-00311-y

DO - 10.1007/s40305-020-00311-y

M3 - Article

AN - SCOPUS:85089312376

SN - 2194-668X

VL - 9

SP - 499

EP - 523

JO - Journal of the Operations Research Society of China

JF - Journal of the Operations Research Society of China

IS - 3

ER -

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

Abstract

Keywords

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