Affine nonexpansive operators, Attouch-Théra duality and the Douglas-Rachford algorithm

2016/03/30 by Bauschke, Heinz H., Lukens, Brett, Moursi, Walaa M. · 1 citation
#47H09 #49M27 #49N15 #90C25 #FOS: Mathematics #Optimization and Control (math.OC) #Primary 47H05 #Secondary 49M29

paper · doi:10.48550/arxiv.1603.09418

Abstract

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-Théra duality. Numerous examples are presented.

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