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Journal of the London Mathematical Society Advance Access published online on April 7, 2009
Journal of the London Mathematical Society, doi:10.1112/jlms/jdn087
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© 2009 London Mathematical Society

Monotone vector fields and the proximal point algorithm on Hadamard manifolds

Chong Li

Department of Mathematics
Zhejiang University
Hangzhou 310027
P.R. China
cli@zju.edu.cn

Genaro López and Victoria Martín-Márquez

Departamento de Análisis Matemático
Universidad de Sevilla
Apdo. 1160, 41080-Sevilla
Spain
victoriam@us.es

The maximal monotonicity notion in Banach spaces is extended to Riemannian manifolds of nonpositive sectional curvature, Hadamard manifolds, and proved to be equivalent to the upper semicontinuity. We consider the problem of finding a singularity of a multivalued vector field in a Hadamard manifold and present a general proximal point method to solve that problem, which extends the known proximal point algorithm in Euclidean spaces. We prove that the sequence generated by our method is well defined and converges to a singularity of a maximal monotone vector field, whenever it exists. Applications in minimization problems with constraints, minimax problems and variational inequality problems, within the framework of Hadamard manifolds, are presented.

2000 Mathematics Subject Classification 47H05, 49J40.

The first author was supported in part by DGES, grant SAB 2006-0195, Spain, and the National Natural Science Foundations of China (grant nos 10671175 and 10731060). The second author was supported by DGES, grant no. MTM2006-13997-C02-01 and Junta de Andalucía, grant no. FQM-127. The third author was supported by Ministerio de Ciencia e Innovación, grant no. AP2005-1018 and Junta de Andalucía, grant no. FQM-127.

Received July 11, 2008; revised November 25, 2008;
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