Skip to main content
SpringerLink
Log in

On an SDP relaxation for kissing number

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

We demonstrate that an earlier semidefinite-programming relaxation for the kissing-number problem cannot provide good upper bounds. Furthermore, we show the existence of an optimal solution for this relaxation that cannot be used as a basis for establishing a good lower bound.

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

Similar content being viewed by others

References

  1. Alizadeh, F., Haeberly, J.-P.A., Overton, M.L.: Complementarity and nondegeneracy in semidefinite programming. Math. Program. 77(1), 111–128 (1997)

    Article  MathSciNet  Google Scholar 

  2. Anstreicher, K.M.: The thirteen spheres: a new proof. Discrete Comput. Geom. 31(4), 613–625 (2004)

    Article  MathSciNet  Google Scholar 

  3. Bachoc, C., Vallentin, F.: New upper bounds for kissing numbers from semidefinite programming. J. Am. Math. Soc. 21, 909–924 (2008)

    Article  MathSciNet  Google Scholar 

  4. Biggs, N.: Algebraic Graph Theory, Cambridge Mathematical Library, 2nd edn. Cambridge University Press, Cambridge (1974)

    Book  Google Scholar 

  5. Boyvalenkov, P., Dodunekov, S., Musin, O.: A survey on the kissing numbers. Serdica Math. J. 38, 507–522 (2012)

    MathSciNet  MATH  Google Scholar 

  6. Conway, J., Sloane, N. (eds.): Sphere Packings. Lattices and Groups. Springer, Berlin (1993)

    Google Scholar 

  7. Jolliffe, I.: Principal Component Analysis, 2nd edn. Springer, Berlin (2010)

    MATH  Google Scholar 

  8. Kucherenko, S., Belotti, P., Liberti, L., Maculan, N.: New formulations for the kissing number problem. Discrete Appl. Math. 155(14), 1837–1841 (2007)

    Article  MathSciNet  Google Scholar 

  9. Liberti, L.: Mathematical programming bounds for kissing numbers. In: Sforza, A., Sterle, C. (eds.) Optimization and Decision Science: Methodologies and Applications, pp. 213–222. Springer, Berlin (2017)

    Chapter  Google Scholar 

  10. Maculan, N., Michelon, P., MacGregor Smith, J.: Bounds on the Kissing Numbers in \({\mathbb{R}}^n\): Mathematical Programming Formulations, Technical report. University of Massachusetts, Amherst (1996)

  11. Mittelmann, H., Vallentin, F.: High-accuracy semidefinite programming bounds for kissing numbers. Exp. Math. 19(2), 175–179 (2010)

    Article  MathSciNet  Google Scholar 

  12. Musin, O.: The kissing number in four dimensions. Ann. Math. 168, 1–32 (2008)

    Article  MathSciNet  Google Scholar 

  13. Pfender, F.: Improved Delsarte bounds for spherical codes in small dimensions. J. Comb. Theory Ser. A 114(6), 1133–1147 (2007)

    Article  MathSciNet  Google Scholar 

  14. Pfender, F., Ziegler, G.: Kissing numbers, sphere packings and some unexpected proofs. Not. Am. Math. Soc. 51(8), 873–883 (2004)

    MathSciNet  MATH  Google Scholar 

  15. Schütte, K., van der Waerden, B.L.: Das Problem der dreizehn Kugeln. Math. Ann. 125, 325–334 (1953)

    Article  MathSciNet  Google Scholar 

  16. Szpiro, G.G.: Kepler’s Conjecture: How Some of the Greatest Minds in History Helped Solve One of the Oldest Math Problems in the World. Wiley, Hoboken (2003)

    MATH  Google Scholar 

Download references

Acknowledgements

J. Lee was supported in part by ONR Grant N00014-17-1-2296. Additionally, part of this work was done while J. Lee was visiting the Simons Institute for the Theory of Computing. It was partially supported by the DIMACS/Simons Collaboration on Bridging Continuous and Discrete Optimization through NSF Grant #CCF-1740425.

Author information

Authors and Affiliations

  1. Department of IOE, University of Michigan, Ann Arbor, MI, USA

    Jon Lee

  2. CNRS LIX, École Polytechnique, 91128, Palaiseau, France

    Leo Liberti

Authors
  1. Jon Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leo Liberti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jon Lee.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lee, J., Liberti, L. On an SDP relaxation for kissing number. Optim Lett 14, 417–422 (2020). https://doi.org/10.1007/s11590-018-1239-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-018-1239-9

Keywords

Search

Navigation