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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Vydáno v:Mathematical methods of operations research (Heidelberg, Germany) Ročník 91; číslo 2; s. 201 - 240
Hlavní autoři: Aragón Artacho, Francisco J., Campoy, Rubén, Tam, Matthew K.
Médium: Journal Article
Jazyk:angličtina
Vydáno: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2020
Springer Nature B.V
Témata:
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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Popis
Shrnutí: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.
Bibliografie:ObjectType-Article-1
SourceType-Scholarly Journals-1
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ISSN:1432-2994
1432-5217
DOI:10.1007/s00186-019-00691-9