Abstract
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.
AMS 2010 Subject Classification: 46B45, 47H10, 90C26
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References
Andres, J., Pastor, K., Šnyrychová: A multivalued version of Sharkovskii’s theorem holds with at most two exceptions. J. Fixed Point Theory Appl. 2, 153–170 (2007)
Andres, J., Fürst, T., Pastor, K.: Full analogy of Sharkovskii’s theorem for lower semicontinuous maps. J. Math. Anal. Appl. 340, 1132–1144 (2008)
Bauschke, H.H., Borwein, J.M.: On projection algorithms for solving convex feasibility problems. SIAM Review 38, 367–426 (1996)
Bauschke, H.H., Combettes, P.L., Luke, D.R.: Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization. J. Opt. Soc. Amer. A 19, 1334–1345 (2002)
Bauschke, H.H., Combettes, P.L., Luke, D.R..: Finding best approximation pairs relative to two closed convex sets in Hilbert spaces. J. Approx. Theory 127, 178–192 (2004)
Bauschke, H.H., Combettes, P.L., Luke, D.R.: A strongly convergent reflection method for finding the projection onto the intersection of two closed convex sets in a Hilbert space. J. Approx. Theory 141, 63–69 (2006)
Douglas, J., Rachford, H.H.: On the numerical solution of heat conduction problems in two or three space variables. Trans. Amer. Math. Soc. 82, 421–439 (1956)
Elser, V., Rankenburg, I., Thibault, P.: Searching with iterated maps. Proceedings of the National Academy of Sciences 104, 418–423 (2007)
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
Lakshmikantham, V., Trigiante, D.: Theory of Difference Equations – Numerical Methods and Applications. Marcel Dekker (2002)
Lions, P.-L., Mercier, B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16, 964–979 (1979)
Pierra, G.: Eclatement de contraintes en parallèle pour la miniminisation d’une forme quadratique. Lecture Notes in Computer Science, Springer, 41 200–218 (1976)
Acknowledgements
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.
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Borwein, J.M., Sims, B. (2011). The Douglas–Rachford Algorithm in the Absence of Convexity. In: Bauschke, H., Burachik, R., Combettes, P., Elser, V., Luke, D., Wolkowicz, H. (eds) Fixed-Point Algorithms for Inverse Problems in Science and Engineering. Springer Optimization and Its Applications(), vol 49. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9569-8_6
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DOI: https://doi.org/10.1007/978-1-4419-9569-8_6
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