On the proximal point method in Hadamard spaces. (3rd October 2017)

• Record Type: Journal Article
• Authors: Chaipunya, Parin; Kumam, Poom
• Is Part Of: Optimization. Volume 66:Number 10(2017)
• Page Start: 1647
• Page End: 1665
• Publication Date: 2017-10-03
• Subjects: Proximal point method -- zero point problem -- optimization -- variational inequality -- Hadamard space -- maximal monotonicity -- Minty's surjectivity condition -- quasilinearization; Mathematical optimization -- Periodicals; 519.7
• DOI: 10.1080/02331934.2017.1349124
• Languages: English
• ISSNs: 0233-1934

Abstract

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.