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The range of the Douglas–Rachford operator in infinite-dimensional Hilbert spaces

Part of: Nonlinear operators and their properties Numerical methods in calculus of variations and optimal control Miscellaneous topics in calculus of variations and optimal control

Published online by Cambridge University Press:  24 October 2025

Walaa M. Moursi
Walaa M. Moursi*
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario N2L 3G1, Canadaand Mansoura University, Mansoura 35516, Egypt
*
e-mail: walaa.moursi@uwaterloo.ca
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Abstract

The Douglas–Rachford algorithm is one of the most prominent splitting algorithms for solving convex optimization problems. Recently, the method has been successful in finding a generalized solution (provided that one exists) for optimization problems in the inconsistent case (i.e., when a solution does not exist). The convergence analysis of the inconsistent case hinges on the study of the range of the displacement operator associated with the Douglas–Rachford splitting operator and the corresponding minimal displacement vector. A comprehensive study of this range has been developed in finite-dimensional Hilbert spaces. In this paper, we provide a formula for the range of the Douglas–Rachford splitting operator in (possibly) infinite-dimensional Hilbert spaces under mild assumptions on the underlying operators. Our new results complement known results in finite-dimensional Hilbert spaces. Several examples illustrate and tighten our conclusions.

Keywords

convex functionconvex setdisplacement mapDouglas–Rachford splitting operatordualitymaximally monotone operatorrangesubdifferential operator

MSC classification

Primary: 47H05: Monotone operators and generalizations 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc. 49M27: Decomposition methods 49N15: Duality theory
Secondary: 90C25: Convex programming 90C46: Optimality conditions, duality 47H14: Perturbations of nonlinear operators 49M29: Methods involving duality

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 18
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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