On the order of the operators in the Douglas-Rachford algorithm
Provides theoretical clarity for practitioners using the Douglas-Rachford algorithm, though the results are incremental.
The paper systematically studies the two possible orderings of operators in the Douglas-Rachford algorithm, showing that the reflectors of the underlying operators act as bijections between the fixed point sets of the two orderings, with elegant formulas under additional assumptions.
The Douglas-Rachford algorithm is a popular method for finding zeros of sums of monotone operators. By its definition, the Douglas-Rachford operator is not symmetric with respect to the order of the two operators. In this paper we provide a systematic study of the two possible Douglas-Rachford operators. We show that the reflectors of the underlying operators act as bijections between the fixed points sets of the two Douglas-Rachford operators. Some elegant formulae arise under additional assumptions. Various examples illustrate our results.