Skip to main content
SpringerLink
Log in

Finding a best approximation pair of points for two polyhedra

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

Given two disjoint convex polyhedra, we look for a best approximation pair relative to them, i.e., a pair of points, one in each polyhedron, attaining the minimum distance between the sets. Cheney and Goldstein showed that alternating projections onto the two sets, starting from an arbitrary point, generate a sequence whose two interlaced subsequences converge to a best approximation pair. We propose a process based on projections onto the half-spaces defining the two polyhedra, which are more negotiable than projections on the polyhedra themselves. A central component in the proposed process is the Halpern–Lions–Wittmann–Bauschke algorithm for approaching the projection of a given point onto a convex set.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

Notes

  1. This acronym was dubbed in [10].

References

  1. Aharoni, R., Duchet, P., Wajnryb, B.: Successive projections on hyperplanes. J. Math. Anal. Appl. 103, 134–138 (1984)

    Article  MathSciNet  Google Scholar 

  2. Bauschke, H.H.: The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space. J. Math. Anal. Appl. 202, 150–159 (1996)

    Article  MathSciNet  Google Scholar 

  3. Bauschke, H.H., Borwein, J.M.: On the convergence of von Neumann’s alternating projection algorithm for two sets. Set-Valued Anal. 1, 185–212 (1993)

    Article  MathSciNet  Google Scholar 

  4. Bauschke, H.H., Borwein, J.M.: Dykstra’s alternating projection algorithm for two sets. J. Approx. Theory 79, 418–443 (1994)

    Article  MathSciNet  Google Scholar 

  5. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. 2nd edn. Springer, Berlin (2017)

  6. Bauschke, H.H., Combettes, P.L., Luke, D.R.: Finding best approximation pairs relative to two closed convex sets in Hilbert spaces. J. Approx. Theory 127, 178–192 (2004)

    Article  MathSciNet  Google Scholar 

  7. 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)

    Article  MathSciNet  Google Scholar 

  8. Byrne, C.L.: Applied Iterative Methods. A K Peters Ltd, Wellesley, MA (2008)

    MATH  Google Scholar 

  9. Cegielski, A.: Iterative Methods for Fixed Point Problems in Hilbert Spaces. Lecture Notes in Mathematics, vol. 2057. Springer, Berlin, Heidelberg (2012)

    MATH  Google Scholar 

  10. Censor, Y.: Computational acceleration of projection algorithms for the linear best approximation problem. Linear Algebra Appl. 416, 111–123 (2006)

    Article  MathSciNet  Google Scholar 

  11. Censor, Y., Cegielski, A.: Projection methods: an annotated bibliography of books and reviews. Optimization 64, 2343–2358 (2015)

    Article  MathSciNet  Google Scholar 

  12. Censor, Y., Chen, W., Combettes, P.L., Davidi, R., Herman, G.T.: On the effectiveness of projection methods for convex feasibility problems with linear inequality constraints. Comput. Optim. Appl. 51, 1065–1088 (2012)

    Article  MathSciNet  Google Scholar 

  13. Censor, Y., Zaknoon, M.: Algorithms and convergence results of projection methods for inconsistent feasibility problems: a review. Pure Appl. Funct. Anal. arXiv:1802.07529

  14. Censor, Y., Zenios, S.A.: Parallel Optimization: Theory, Algorithms, and Applications. Oxford University Press, New York (1997)

    MATH  Google Scholar 

  15. Cheney, W., Goldstein, A.A.: Proximity maps for convex sets. Proc. Am. Math. Soc. 10, 448–450 (1959)

    Article  MathSciNet  Google Scholar 

  16. Deutsch, F.: Best Approximation in Inner Product Spaces. Springer, New York (2001)

    Chapter  Google Scholar 

  17. Dykstra, R.L.: An algorithm for restricted least squares regression. J. Am. Stat. Assoc. 78, 837–842 (1983)

    Article  MathSciNet  Google Scholar 

  18. Escalante, R., Raydan, M.: Alternating Projection Methods. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2011)

    Book  Google Scholar 

  19. Galántai, A.: Projectors and Projection Methods. Kluwer Academic Publishers, Dordrecht (2004)

    Book  Google Scholar 

  20. Halpern, B.: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 73, 957–961 (1967)

    Article  MathSciNet  Google Scholar 

  21. Kopecká, E., Reich, S.: A note on alternating projections in Hilbert space. J. Fixed Point Theory Appl. 12, 41–47 (2012)

    Article  MathSciNet  Google Scholar 

  22. Lions, P.L.: Approximation de points fixes de contractions. C. R. Acad. Sci. Paris Sér. A-B 284, A1357–A1359 (1977)

    MathSciNet  Google Scholar 

  23. Luke, D.R.: Finding best approximation pairs relative to a convex and prox-regular set in a Hilbert space. SIAM J. Optim. 19, 714–739 (2008)

    Article  MathSciNet  Google Scholar 

  24. Meshulam, R.: On products of projections. Discrete Math. 154, 307–310 (1996)

    Article  MathSciNet  Google Scholar 

  25. Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. (Basel) 58, 486–491 (1992)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank Yehuda Zur for Matlab programming work at the early stages of our research.

Author information

Authors and Affiliations

  1. Department of Mathematics, Technion – Israel Institute of Technology, Technion City, Haifa, 3200003, Israel

    Ron Aharoni & Zilin Jiang

  2. Department of Mathematics, University of Haifa, Mt. Carmel, Haifa, 3498838, Israel

    Yair Censor

Authors
  1. Ron Aharoni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yair Censor
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zilin Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zilin Jiang.

Additional information

Ron Aharoni: Supported in part by the United States–Israel Binational Science Foundation (BSF) Grant No. 2012031, the Israel Science Foundation (ISF) Grant No. 2023464 and the Discount Bank Chair at the Technion. Yair Censor: Supported in part by BSF Grant No. 2013003. Zilin Jiang: Supported in part by ISF Grant Nos. 1162/15, 936/16.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Aharoni, R., Censor, Y. & Jiang, Z. Finding a best approximation pair of points for two polyhedra. Comput Optim Appl 71, 509–523 (2018). https://doi.org/10.1007/s10589-018-0021-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-018-0021-3

Keywords

Mathematics Subject Classification

Search

Navigation