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

In: Splitting Algorithms, Modern Operator Theory, and Applications

Author

Listed:
  • Heinz H. Bauschke

    (University of British Columbia, Department of Mathematics)

  • Sylvain Gretchko

    (Mathematics, UBCO)

  • Walaa M. Moursi

    (Stanford University, Electrical Engineering
    Mansoura University, Faculty of Science, Mathematics Department)

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.

Suggested Citation

  • Heinz H. Bauschke & Sylvain Gretchko & Walaa M. Moursi, 2019. "Numerical Explorations of Feasibility Algorithms for Finding Points in the Intersection of Finite Sets," Springer Books, in: Heinz H. Bauschke & Regina S. Burachik & D. Russell Luke (ed.), Splitting Algorithms, Modern Operator Theory, and Applications, chapter 0, pages 69-90, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-25939-6_3
    DOI: 10.1007/978-3-030-25939-6_3
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