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The Douglas–Rachford Algorithm in the Absence of Convexity

In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering

Author

Listed:
  • Jonathan M. Borwein

    (University of Newcastle)

  • Brailey Sims

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.

Suggested Citation

  • Jonathan M. Borwein & Brailey Sims, 2011. "The Douglas–Rachford Algorithm in the Absence of Convexity," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 93-109, Springer.
    • Handle: RePEc:spr:spochp:978-1-4419-9569-8_6
    DOI: 10.1007/978-1-4419-9569-8_6
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