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

TL;DR

This paper characterizes the range of the Douglas-Rachford operator in infinite-dimensional Hilbert spaces, providing new formulas that extend finite-dimensional results and aid in understanding the algorithm's behavior in inconsistent convex optimization problems.

Contribution

It offers a new formula for the range of the Douglas-Rachford operator in infinite-dimensional spaces, enhancing theoretical understanding of the method's convergence properties.

Findings

Derived a formula for the operator's range in infinite-dimensional spaces

Extended finite-dimensional results to infinite-dimensional settings

Provided examples illustrating the theoretical conclusions

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.