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arXiv · 1603.09418

Affine nonexpansive operators, Attouch-Th\'era duality and the Douglas-Rachford algorithm

Heinz H. Bauschke·Brett Lukens·Walaa M. Moursi
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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-Th\'era duality. Numerous examples are presented.

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Heinz H. Bauschke, Brett Lukens, Walaa M. Moursi. 2016-03-30. Affine nonexpansive operators, Attouch-Th\'era duality and the Douglas-Rachford algorithm. https://arxiv.org/abs/1603.09418

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