The Douglas–Rachford algorithm for convex and nonconvex feasibility problems

The Douglas–Rachford algorithm is an optimization method that can be used for solving feasibility problems. To apply the method, it is necessary that the problem at hand is prescribed in terms of constraint sets having efficiently computable nearest points. Although the convergence of the algorithm...

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Vydané v:	Mathematical methods of operations research (Heidelberg, Germany) Ročník 91; číslo 2; s. 201 - 240
Hlavní autori:	Aragón Artacho, Francisco J., Campoy, Rubén, Tam, Matthew K.
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2020 Springer Nature B.V
Predmet:	Algorithms Business and Management Calculus of Variations and Optimal Control; Optimization Combinatorial analysis Convergence Feasibility Mathematics Mathematics and Statistics Operations research Operations Research/Decision Theory Optimization Original Article 65K05 Projection methods Douglas–Rachford Eight queens problem 90C27 Feasibility problem 90-01 65-01
ISSN:	1432-2994, 1432-5217
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Abstract	The Douglas–Rachford algorithm is an optimization method that can be used for solving feasibility problems. To apply the method, it is necessary that the problem at hand is prescribed in terms of constraint sets having efficiently computable nearest points. Although the convergence of the algorithm is guaranteed in the convex setting, the scheme has demonstrated to be a successful heuristic for solving combinatorial problems of different type. In this self-contained tutorial, we develop the convergence theory of projection algorithms within the framework of fixed point iterations, explain how to devise useful feasibility problem formulations, and demonstrate the application of the Douglas–Rachford method to said formulations. The paradigm is then illustrated on two concrete problems: a generalization of the “eight queens puzzle” known as the “( m , n )-queens problem”, and the problem of constructing a probability distribution with prescribed moments.
AbstractList	The Douglas–Rachford algorithm is an optimization method that can be used for solving feasibility problems. To apply the method, it is necessary that the problem at hand is prescribed in terms of constraint sets having efficiently computable nearest points. Although the convergence of the algorithm is guaranteed in the convex setting, the scheme has demonstrated to be a successful heuristic for solving combinatorial problems of different type. In this self-contained tutorial, we develop the convergence theory of projection algorithms within the framework of fixed point iterations, explain how to devise useful feasibility problem formulations, and demonstrate the application of the Douglas–Rachford method to said formulations. The paradigm is then illustrated on two concrete problems: a generalization of the “eight queens puzzle” known as the “(m, n)-queens problem”, and the problem of constructing a probability distribution with prescribed moments. The Douglas–Rachford algorithm is an optimization method that can be used for solving feasibility problems. To apply the method, it is necessary that the problem at hand is prescribed in terms of constraint sets having efficiently computable nearest points. Although the convergence of the algorithm is guaranteed in the convex setting, the scheme has demonstrated to be a successful heuristic for solving combinatorial problems of different type. In this self-contained tutorial, we develop the convergence theory of projection algorithms within the framework of fixed point iterations, explain how to devise useful feasibility problem formulations, and demonstrate the application of the Douglas–Rachford method to said formulations. The paradigm is then illustrated on two concrete problems: a generalization of the “eight queens puzzle” known as the “( m , n )-queens problem”, and the problem of constructing a probability distribution with prescribed moments.
Author	Campoy, Rubén Aragón Artacho, Francisco J. Tam, Matthew K.
Author_xml	– sequence: 1 givenname: Francisco J. orcidid: 0000-0002-2445-8011 surname: Aragón Artacho fullname: Aragón Artacho, Francisco J. email: francisco.aragon@ua.es organization: Department of Mathematics, University of Alicante – sequence: 2 givenname: Rubén surname: Campoy fullname: Campoy, Rubén organization: Department of Mathematics, University of Alicante – sequence: 3 givenname: Matthew K. surname: Tam fullname: Tam, Matthew K. organization: Institute for Numerical and Applied Mathematics, University of Göttingen
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Snippet	The Douglas–Rachford algorithm is an optimization method that can be used for solving feasibility problems. To apply the method, it is necessary that the...
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SubjectTerms	Algorithms Business and Management Calculus of Variations and Optimal Control; Optimization Combinatorial analysis Convergence Feasibility Mathematics Mathematics and Statistics Operations research Operations Research/Decision Theory Optimization Original Article
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Title	The Douglas–Rachford algorithm for convex and nonconvex feasibility problems
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