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Linear convergence of the generalized Douglas–Rachford algorithm for feasibility problems

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Abstract

In this paper, we study the generalized Douglas–Rachford algorithm and its cyclic variants which include many projection-type methods such as the classical Douglas–Rachford algorithm and the alternating projection algorithm. Specifically, we establish several local linear convergence results for the algorithm in solving feasibility problems with finitely many closed possibly nonconvex sets under different assumptions. Our findings not only relax some regularity conditions but also improve linear convergence rates in the literature. In the presence of convexity, the linear convergence is global.

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Acknowledgements

The authors are thankful to the editors and the referees for their constructive comments. MND was partially supported by the Australian Research Council (ARC) under Discovery Project 160101537 and by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant 101.02-2016.11. This work was essentially completed during MND’s visit to UMass Lowell in September 2017 to whom he acknowledges the hospitality.

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Authors and Affiliations

  1. CARMA, University of Newcastle, Callaghan, NSW, 2308, Australia

    Minh N. Dao

  2. Department of Mathematical Sciences, Kennedy College of Sciences, University of Massachusetts Lowell, Lowell, MA, 01854, USA

    Hung M. Phan

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  1. Minh N. Dao
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  2. Hung M. Phan
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Correspondence to Minh N. Dao.

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Dao, M.N., Phan, H.M. Linear convergence of the generalized Douglas–Rachford algorithm for feasibility problems. J Glob Optim 72, 443–474 (2018). https://doi.org/10.1007/s10898-018-0654-x

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