Affine Nonexpansive Operators, Attouch-Thera Duality and the Douglas-Rachford Algorithm

作     者：Bauschke, Heinz H. Lukens, Brett Moursi, Walaa M. 

作者机构：Univ British Columbia Math Kelowna BC V1V 1V7 Canada 3990 Lansdowne Rd Armstrong BC V0E 1B3 Canada Mansoura Univ Math Dept Fac Sci Mansoura 35516 Egypt 

出 版 物：《SET-VALUED AND VARIATIONAL ANALYSIS》 (集值分析)

年 卷 期：2017年第25卷第3期

页      面：481-505页

核心收录：

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：Natural Sciences and Engineering Research Council of Canada Canada Research Chair Program 

主　　题：Affine mapping Attouch-Thera duality Douglas-Rachford algorithm Linear convergence Maximally monotone operator Nonexpansive mapping Paramonotone operator Strong convergence Toeplitz matrix Tridiagonal matrix 

摘      要：The Douglas-Rachford splitting algorithm was originally proposed in 1956 to solve a system of linear equations arising from the discretization of a partial differential equation. In 1979, Lions and Mercier brought forward a very powerful extension of this method suitable to solve optimization problems. In this paper, we revisit the original affine setting. We provide a powerful convergence result for finding a zero of the sum of two maximally monotone affine relations. As a by product of our analysis, we obtain results concerning the convergence of iterates of affine nonexpansive mappings as well as Attouch-Th,ra duality. Numerous examples are presented.