A Lyapunov-type approach to convergence of the Douglas–Rachford algorithm for a nonconvex setting
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The Douglas–Rachford projection algorithm is an iterative method used to find a point in the intersection of closed constraint sets. The algorithm has been experimentally observed to solve various nonconvex feasibility problems; an observation which current theory cannot sufficiently explain. In this paper, we prove convergence of the Douglas–Rachford algorithm in a potentially nonconvex setting. Our analysis relies on the existence of a Lyapunov-type functional whose convexity properties are not tantamount to convexity of the original constraint sets. Moreover, we provide various nonconvex examples in which our framework proves global convergence of the algorithm.
KeywordsDouglas–Rachford algorithm Feasibility problem Global convergence Graph of a function Linear convergence Lyapunov function Method of alternating projections Newton’s method Nonconvex set Projection Stability Zero of a function
Mathematics Subject Classification90C26 47H10 37B25
This paper is dedicated to the memory of Jonathan M. Borwein and his enthusiasm for the Douglas–Rachford algorithm. MND was partially supported by the Australian Research Council Discovery Project DP160101537 and a Startup Research Grant from the University of Newcastle. He wishes to acknowledge the hospitality and the support of D. Russell Luke during his visit to Universität Göttingen. MKT was partially supported by the Deutsche Forschungsgemeinschaft RTG 2088 and a Postdoctoral Fellowship from the Alexander von Humboldt Foundation. The authors wish to thank the three anonymous referees for their comments and suggestions.
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