The Douglas–Rachford Algorithm in the Absence of Convexity
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.
KeywordsNon-convex feasibility problem Fixed point theory Dynamical system Iteration
Unable to display preview. Download preview PDF.
This research was supported by the Australian Research Council. We also express our thanks to Chris Maitland, Matt Skerritt and Ulli Kortenkamp for helping us exploit the full resources of Cinderella.
- 9.Gravel, S., Elser, V.: Divide and concur: A general approach constraint satisfaction. Phys. Rev. E 78 036706, pp. 5 (2008), http://link.aps.org/doi/10.1103/PhysRevE.78.036706
- 10.Lakshmikantham, V., Trigiante, D.: Theory of Difference Equations – Numerical Methods and Applications. Marcel Dekker (2002)Google Scholar
- 12.Pierra, G.: Eclatement de contraintes en parallèle pour la miniminisation d’une forme quadratique. Lecture Notes in Computer Science, Springer, 41 200–218 (1976)Google Scholar