Abstract
We expand upon previous work that examined the behavior of the iterated Douglas-Rachford method for a line and a circle by considering two generalizations:that of a line and an ellipse and that of a line together with a p-sphere. With computer assistance we discover a beautiful geometry that illustrates phenomena which may affect the behavior of the iterates by slowing or inhibiting convergence for feasible cases. We prove local convergence near feasible points, and—seeking a better understanding of the behavior—we employ parallelization in order to study behavior graphically. Motivated by the computer-assisted discoveries, we prove a result about behavior of the method in infeasible cases.
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The authors wish to especially thank one of the referees for their careful reading of the manuscript and their several suggestions which have significantly improved the readability of the final version.
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This work is dedicated to the memory of Jonathan M. Borwein our greatly missed friend, mentor, and colleague. His early contributions to this paper, like his contributions to so much else, were both vital and inspirational.
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Borwein, J.M., Lindstrom, S.B., Sims, B. et al. Dynamics of the Douglas-Rachford Method for Ellipses and p-Spheres. Set-Valued Var. Anal 26, 385–403 (2018). https://doi.org/10.1007/s11228-017-0457-0
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DOI: https://doi.org/10.1007/s11228-017-0457-0