Numerical Explorations of Feasibility Algorithms for Finding Points in the Intersection of Finite Sets
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.
KeywordsCyclic Douglas–Rachford algorithm Douglas–Rachford algorithm Extrapolated parallel projection method Method of cyclic projections Nonconvex feasibility problem Optimization algorithm Projection
AMS 2010 Subject Classification49M20 49M27 49M37 65K05 65K10 90C25 90C26 90C30
We thank the referee for constructive comments and suggestions. The research of HHB was partially supported by NSERC.
- 4.Bauschke, H.H., Koch, V.R.: Projection methods: Swiss army knives for solving feasibility and best approximation problems with halfspaces. Contemp. Math. 636, 1–40 (2015) doi: 10.1090/conm/636/12726Google Scholar
- 6.Bauschke, H.H., Moursi, W.M.: On the Douglas–Rachford algorithm. Math. Prog. (Ser. A) 164, 263–284 (2017)Google Scholar
- 12.Censor, Y., Zaknoon, M.: Algorithms and convergence results of projection methods for inconsistent feasibility problems: a review. Pure Appl. Funct. Anal. 3, 565–586 (2018) https://arxiv.org/abs/1802.07529 [math.OC] (2018)
- 13.Censor, Y., Zenios, S.A.: Parallel Optimization. Oxford University Press (1997)Google Scholar
- 17.Eckstein, J., Bertsekas, D.P.: On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Prog. (Ser. A) 55, 293–318 (1992)Google Scholar
- 21.Luke, D.R., Sabach, S., Teboulle, M.: Optimization on spheres: models and proximal algorithms with computational performance comparisons. https://arxiv.org/abs/1810.02893 [math.OC] (2018)