A Cyclic Douglas–Rachford Iteration Scheme



In this paper, we present two Douglas–Rachford inspired iteration schemes which can be applied directly to N-set convex feasibility problems in Hilbert space. Our main results are weak convergence of the methods to a point whose nearest point projections onto each of the N sets coincide. For affine subspaces, convergence is in norm. Initial results from numerical experiments, comparing our methods to the classical (product-space) Douglas–Rachford scheme, are promising.


Douglas–Rachford method Convex feasibility problem Projections Firmly nonexpansive map Nonexpansive map Asymptotic regularity Fixed points Parallelization 



The authors wish to acknowledge Francisco J. Aragón Artacho, Brailey Sims, Simeon Reich, and the two anonymous referees for their helpful comments and suggestions.

Jonathan M. Borwein’s research is supported in part by the Australian Research Council.

Matthew K. Tam’s research is supported in part by an Australian Postgraduate Award.


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© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.CARMA CentreUniversity of NewcastleCallaghanAustralia

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