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Douglas-Rachford Method for Non-Convex Feasibility Problems

Banff International Research Station for Mathematical Innovation and Discovery

The Douglas-Rachford method has been employed successfully to solve a variety of non-convex feasibility problems. In particular, it shows surprising stability when applied to finding the intersections of hypersurfaces. We prove local convergence in the generalization of a case prototypical of the phase retrieval problem. In so doing, we also discover phenomena which may inhibit convergence. Finally we illustrate an application to solving boundary valued ordinary differential equations.
This talk includes discoveries from three closely related works:
1. With Brailey Sims, Matthew Skerritt. ''Computing Intersections of Implicitly Specified Plane Curves.'' To appear in <em>Journal of Nonlinear and Convex Analysis</em>.
2. With Jonathan M. Borwein, Brailey Sims, Anna Schneider, Matthew Skerritt. ''Dynamics of the Douglas-Rachford Method for Ellipses and p-Spheres.'' Submitted to <em>Set Valued and Variational Analysis</em>.
3. With Bishnu Lamichhane and Brailey Sims. ''Application of Projection Algorithms to Differential Equations: Boundary Value Problems,'' in preparation with plans to submit to <em>ANZIAM Journal</em>.

Mathematics; Operations research, mathematical programming; Operator theory; Control/optimization/operation research

video/mp4

Author affiliation: University of Newcastle

2018-03-24

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/64910

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/