A Lyapunov-type approach to convergence of the Douglas–Rachford algorithm for a nonconvex setting

Minh N. Dao () and Matthew K. Tam ()
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Minh N. Dao: University of Newcastle
Matthew K. Tam: Universität Göttingen

Journal of Global Optimization, 2019, vol. 73, issue 1, No 4, 83-112

Abstract: Abstract The Douglas–Rachford projection algorithm is an iterative method used to find a point in the intersection of closed constraint sets. The algorithm has been experimentally observed to solve various nonconvex feasibility problems; an observation which current theory cannot sufficiently explain. In this paper, we prove convergence of the Douglas–Rachford algorithm in a potentially nonconvex setting. Our analysis relies on the existence of a Lyapunov-type functional whose convexity properties are not tantamount to convexity of the original constraint sets. Moreover, we provide various nonconvex examples in which our framework proves global convergence of the algorithm.

Keywords: Douglas–Rachford algorithm; Feasibility problem; Global convergence; Graph of a function; Linear convergence; Lyapunov function; Method of alternating projections; Newton’s method; Nonconvex set; Projection; Stability; Zero of a function; 90C26; 47H10; 37B25 (search for similar items in EconPapers)
Date: 2019
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10898-018-0677-3

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