On the range of the Douglas-Rachford operator
This work addresses a fundamental gap in understanding the Douglas-Rachford operator's range, providing theoretical insights for convex optimization and monotone operator theory.
The paper systematically studies the range of the Douglas-Rachford operator for 3* monotone operators, deriving a formula showing the range is nearly equal to a simple set involving the domains and ranges of the underlying operators. Similar results are obtained for the displacement mapping, with applications to subdifferential operators and the Brezis-Haraux theorem.
The problem of finding a minimizer of the sum of two convex functions - or, more generally, that of finding a zero of the sum of two maximally monotone operators - is of central importance in variational analysis. Perhaps the most popular method of solving this problem is the Douglas-Rachford splitting method. Surprisingly, little is known about the range of the Douglas-Rachford operator. In this paper, we set out to study this range systematically. We prove that for 3* monotone operators a very pleasing formula can be found that reveals the range to be nearly equal to a simple set involving the domains and ranges of the underlying operators. A similar formula holds for the range of the corresponding displacement mapping. We discuss applications to subdifferential operators, to the infimal displacement vector, and to firmly nonexpansive mappings. Various examples and counter-examples are presented, including some concerning the celebrated Brezis-Haraux theorem.