On the Douglas-Rachford algorithm
This solves a decade-old open problem for researchers working on splitting algorithms and convex optimization, providing a complete theoretical understanding of the algorithm's behavior in the inconsistent case.
The paper resolves the long-standing open problem of weak convergence of the Douglas-Rachford algorithm in the inconsistent convex feasibility setting, proving that the shadow sequence converges weakly to a solution of a best approximation problem. A more general sufficient condition for weak convergence in the general inconsistent case is also provided.
The Douglas-Rachford algorithm is a very popular splitting technique for finding a zero of the sum of two maximally monotone operators. However, the behaviour of the algorithm remains mysterious in the general inconsistent case, i.e., when the sum problem has no zeros. More than a decade ago, however, it was shown that in the (possibly inconsistent) convex feasibility setting, the shadow sequence remains bounded and it is weak cluster points solve a best approximation problem. In this paper, we advance the understanding of the inconsistent case significantly by providing a complete proof of the full weak convergence in the convex feasibility setting. In fact, a more general sufficient condition for the weak convergence in the general case is presented. Several examples illustrate the results.