Global behavior of the Douglas–Rachford method for a nonconvex feasibility problem

In recent times the Douglas–Rachford algorithm has been observed empirically to solve a variety of nonconvex feasibility problems including those of a combinatorial nature. For many of these problems current theory is not sufficient to explain this observed success and is mainly concerned with questions of local convergence. In this paper we analyze global behavior of the method for finding a point in the intersection of a half-space and a potentially non-convex set which is assumed to satisfy a well-quasi-ordering property or a property weaker than compactness. In particular, the special case in which the second set is finite is covered by our framework and provides a prototypical setting for combinatorial optimization problems.

F.J. Aragón Artacho was supported by MINECO of Spain and FEDER of EU, as part of the Ramón y Cajal program (RYC-2013-13327) and the Grant MTM2014-59179-C2-1-P. J.M. Borwein was supported, in part, by the Australian Research Council. M.K. Tam was supported by an Australian Post-Graduate Award.