The Douglas–Rachford algorithm for a hyperplane and a doubleton
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The Douglas–Rachford algorithm is a popular algorithm for solving both convex and nonconvex feasibility problems. While its behaviour is settled in the convex inconsistent case, the general nonconvex inconsistent case is far from being fully understood. In this paper, we focus on the most simple nonconvex inconsistent case: when one set is a hyperplane and the other a doubleton (i.e., a two-point set). We present a characterization of cycling in this case which—somewhat surprisingly—depends on whether the ratio of the distance of the points to the hyperplane is rational or not. Furthermore, we provide closed-form expressions as well as several concrete examples which illustrate the dynamical richness of this algorithm.
KeywordsClosed-form expressions Cycling Douglas–Rachford algorithm Feasibility problem Finite set Hyperplane Method of alternating projections Projector Reflector
Mathematics Subject ClassificationPrimary 47H10 49M27 Secondary 65K05 65K10 90C26
The authors thank two anonymous referees for their careful reading and constructive comments. HHB was partially supported by the Natural Sciences and Engineering Research Council of Canada (Grant No. 2018-03703). MND was partially supported by the Australian Research Council (Grant No. DP160101537).
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