A Cyclic Douglas–Rachford Iteration Scheme

Borwein, Jonathan M.; Tam, Matthew K.

doi:10.1007/s10957-013-0381-x

Cited by 65 publications

(89 citation statements)

References 36 publications

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“…In general, von Neumann's alternating projection is unable to find a point in the intersection of H and Q (and hence the same is true for the cyclic Douglas-Rachford algorithm [5,6]). Figure 2 shows a simple example with a doubleton Q = {q 1 , q 2 } ⊂ R 2 .…”

Section: Preliminariesmentioning

confidence: 99%

Global behavior of the Douglas–Rachford method for a nonconvex feasibility problem

2015

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In recent times the Douglas-Rachford algorithm has been observed empirically to solve a variety of nonconvex feasibility problems including those of a combinatorial nature. For many of these problems current theory is not sufficient to explain this observed success and is mainly concerned with questions of local convergence. In this paper we analyze global behavior of the method for finding a point in the intersection of a half-space and a potentially non-convex set which is assumed to satisfy a wellquasi-ordering property or a property weaker than compactness. In particular, the special case in which the second set is finite is covered by our framework and provides a prototypical setting for combinatorial optimization problems.

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Section: Preliminariesmentioning

confidence: 99%

Global behavior of the Douglas–Rachford method for a nonconvex feasibility problem

2015

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“…The drawback with this approach is that minimizing φ is usually intractable as its Lipschitz constant cannot be estimated or, even worst, it is not Lipschitz continuous (unless one can replace R n with some polytope). Another potential approach consists of extending to infinitely many sets (in this case the half-spaces { x ∈ R n : a (t) ⊤ x ≥ b (t) } , t ∈ T ) the Douglas-Rachford method for finite families of closed convex sets [15], but proving the convergence could be a hard task.…”

Section: Introductionmentioning

confidence: 99%

A comparative note on the relaxation algorithms for the linear semi-infinite feasibility problem

et al. 2016

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The problem (LF P ) of finding a feasible solution to a given linear semi-infinite system arises in different contexts. This paper provides an empirical comparative study of relaxation algorithms for (LF P ). In this study we consider, together with the classical algorithm, implemented with different values of the fixed parameter (the step size), a new relaxation algorithm with random parameter which outperforms the classical one in most test problems whatever fixed parameter is taken. This new algorithm converges geometrically to a feasible solution under mild conditions. The relaxation algorithms under comparison have been implemented using the Extended Cutting Angle Method (ECAM) for solving the global optimization subproblems.

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“…See for example, the method of cyclic projections [5,8], Dykstra's method [19,6,13], the cyclic Douglas-Rachford method [17], and many references contained in these papers.…”

Section: Introductionmentioning

confidence: 99%