On the proximal point method in Hadamard spaces
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Publication Details
Author list: Chaipunya P., Kumam P.
Publisher: Taylor and Francis Group
Publication year: 2017
Journal: Optimization (0233-1934)
Volume number: 66
Issue number: 10
Start page: 1647
End page: 1665
Number of pages: 19
ISSN: 0233-1934
eISSN: 1029-4945
Languages: English-Great Britain (EN-GB)
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Abstract
Proximal point method is one of the most influential procedure in solving nonlinear variational problems. It has recently been introduced in Hadamard spaces for solving convex optimization, and later for variational inequalities. In this paper, we study the general proximal point method for finding a zero point of a maximal monotone set-valued vector field defined on a Hadamard space and valued in its dual. We also give the relation between the maximality and Minty’s surjectivity condition, which is essential for the proximal point method to be well-defined. By exploring the properties of monotonicity and the surjectivity condition, we were able to show under mild assumptions that the proximal point method converges weakly to a zero point. Additionally, by taking into account the metric subregularity, we obtained the local strong convergence in linear and super-linear rates. © 2017 Informa UK Limited, trading as Taylor & Francis Group.
Keywords
Hadamard space, maximal monotonicity, Minty’s surjectivity condition, zero point problem
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