The Douglas–Rachford Algorithm in the Absence of Convexity

Part of the Springer Optimization and Its Applications book series (SOIA, volume 49)


The Douglas–Rachford iteration scheme, introduced half a century ago in connection with nonlinear heat flow problems, aims to find a point common to two or more closed constraint sets. Convergence of the scheme is ensured when the sets are convex subsets of a Hilbert space, however, despite the absence of satisfactory theoretical justification, the scheme has been routinely used to successfully solve a diversity of practical problems in which one or more of the constraints involved is non-convex. As a first step toward addressing this deficiency, we provide convergence results for a prototypical non-convex two-set scenario in which one of the sets is the Euclidean sphere.


Non-convex feasibility problem Fixed point theory Dynamical system Iteration 


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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.CARMA, School of Mathematical and Physical SciencesUniversity of NewcastleNewcastleAustralia

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