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arXiv · 2206.07204

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

Walaa M. Moursi
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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. In this paper, we provide a formula for the range of the Douglas-Rachford splitting operator in (possibly) infinite-dimensional Hilbert space 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.

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Walaa M. Moursi. 2022-06-14. The range of the Douglas-Rachford operator in infinite-dimensional Hilbert spaces. https://arxiv.org/abs/2206.07204

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