Journal of Global Optimization

, Volume 57, Issue 3, pp 753–769 | Cite as

Global convergence of a non-convex Douglas–Rachford iteration

  • Francisco J. Aragón ArtachoEmail author
  • Jonathan M. Borwein


We establish a region of convergence for the proto-typical non-convex Douglas–Rachford iteration which finds a point on the intersection of a line and a circle. Previous work on the non-convex iteration Borwein and Sims (Fixed-point algorithms for inverse problems in science and engineering, pp. 93–109, 2011) was only able to establish local convergence, and was ineffective in that no explicit region of convergence could be given.


Non-convex feasibility problem Fixed point theory Projection algorithm Douglas–Rachford algorithm Global convergence Signal reconstruction 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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. Am. A 19(7), 1334–1345 (2002)CrossRefGoogle Scholar
  2. 2.
    Borwein, J.M., Sims, B.: The Douglas–Rachford Algorithm in the Absence of Convexity. In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Springer Optimization and its Applications 49, pp. 93–109 (2011)Google Scholar
  3. 3.
    Borwein J.M, Zhu Q.J: Techniques of Variational Analysis. Springer, New York (2005)Google Scholar
  4. 4.
    Elser V., Rankenburg I., Thibault P.: Searching with iterated maps. Proc. Natl. Acad. Sci. 104, 418–423 (2007)CrossRefGoogle Scholar
  5. 5.
    Goebel K., Kirk W.A: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  6. 6.
    Gravel S., Elser V.: Divide and concur: a general approach constraint satisfaction. Phys. Rev. E 78(036706), 1–5 (2008)Google Scholar
  7. 7.
    Lions P.L, Mercier B.: Splitting algorithms for the sum of two nonlinear operators. SIAM J. Numer. Anal. 16(6), 964–979 (1979)CrossRefGoogle Scholar
  8. 8.
    Opial Z.: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 73, 591–597 (1967)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2012

Authors and Affiliations

  • Francisco J. Aragón Artacho
    • 1
    Email author
  • Jonathan M. Borwein
    • 1
    • 2
  1. 1.Centre for Computer Assisted Research Mathematics and its Applications (CARMA)University of NewcastleCallaghanAustralia
  2. 2.King Abdul-Aziz UniversityJeddahSaudi Arabia

Personalised recommendations