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Numerical Explorations of Feasibility Algorithms for Finding Points in the Intersection of Finite Sets

  • Heinz H. BauschkeEmail author
  • Sylvain Gretchko
  • Walaa M. Moursi
Chapter

Abstract

Projection methods are popular algorithms for iteratively solving feasibility problems in Euclidean or even Hilbert spaces. They employ (selections of) nearest point mappings to generate sequences that are designed to approximate a point in the intersection of a collection of constraint sets. Theoretical properties of projection methods are fairly well understood when the underlying constraint sets are convex; however, convergence results for the nonconvex case are more complicated and typically only local. In this paper, we explore the perhaps simplest instance of a feasibility algorithm, namely when each constraint set consists of only finitely many points. We numerically investigate four constellations: either few or many constraint sets, with either few or many points. Each constellation is tackled by four popular projection methods each of which features a tuning parameter. We examine the behaviour for a single and for a multitude of orbits, and we also consider local and global behaviour. Our findings demonstrate the importance of the choice of the algorithm and that of the tuning parameter.

Keywords

Cyclic Douglas–Rachford algorithm Douglas–Rachford algorithm Extrapolated parallel projection method Method of cyclic projections Nonconvex feasibility problem Optimization algorithm Projection 

AMS 2010 Subject Classification

49M20 49M27 49M37 65K05 65K10 90C25 90C26 90C30 

Notes

Acknowledgements

We thank the referee for constructive comments and suggestions. The research of HHB was partially supported by NSERC.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Heinz H. Bauschke
    • 1
    Email author
  • Sylvain Gretchko
    • 2
  • Walaa M. Moursi
    • 3
    • 4
  1. 1.Department of MathematicsUniversity of British ColumbiaKelownaCanada
  2. 2.Mathematics, UBCOKelownaCanada
  3. 3.Electrical EngineeringStanford UniversityStanfordUSA
  4. 4.Faculty of Science, Mathematics DepartmentMansoura UniversityMansouraEgypt

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