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Monotone vector fields and the proximal point algorithm on Hadamard manifolds
Author(s) -
Li Chong,
López Genaro,
Martín-Márquez Victoria
Publication year - 2009
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/jdn087
Subject(s) - mathematics , hadamard transform , hadamard three lines theorem , variational inequality , singularity , monotone polygon , monotonic function , pure mathematics , sequence (biology) , banach space , manifold (fluid mechanics) , mathematical analysis , hadamard product , geometry , mechanical engineering , biology , engineering , genetics
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.