Dynamics of the Douglas-Rachford Method for Ellipses and p-Spheres

• Published in: Set-valued and variational analysis Vol. 26; no. 2; pp. 385 - 403
• Main Authors: Borwein, Jonathan M., Lindstrom, Scott B., Sims, Brailey, Schneider, Anna, Skerritt, Matthew P.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01-06-2018; Springer Nature B.V
• Subjects: Analysis; Convergence; Mathematics; Mathematics and Statistics; Optimization; Parallel processing; Douglas-Rachford; 49M30; 47H99; Feasibility; 90C26; Iterative methods; Projection algorithms; Discrete dynamical systems; 65Q30
• ISSN: 1877-0533; 1877-0541
• DOI: 10.1007/s11228-017-0457-0

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.