The range of the Douglas-Rachford operator in infinite-dimensional Hilbert spaces
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.
Keywords
Cite
@article{arxiv.2206.07204,
title = {The range of the Douglas-Rachford operator in infinite-dimensional Hilbert spaces},
author = {Walaa M. Moursi},
journal= {arXiv preprint arXiv:2206.07204},
year = {2022}
}