Skip to main content
Log in

On the Problem of Packing Spheres in a Cube

  • Published:
Cybernetics and Systems Analysis Aims and scope

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

References

  1. D. Gilbert and S. Cohn-Vossen, Anschauliche Geometrie [Russian translation], 3d Edition, Nauka, Moscow (1981).

    Google Scholar 

  2. Neil J. A. Sloane, “The Packing of Spheres,” Scientific American , 250 (January, 1984), No. 3, 116–125 (1984).

    Article  Google Scholar 

  3. J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices, and Groups, 2nd Edition, Springer, New York (1993).

    Book  MATH  Google Scholar 

  4. N. Z. Shor and S. I. Stetsenko, Quadratic Extremal Problems and Nondifferentiable Optimization [in Russian], Naukova Dumka, Kyiv (1989).

    Google Scholar 

  5. N. Z. Shor, Nondifferentiable Optimization and Polynomial Problems, Kluwer, Dordrecht (1998).

    Book  MATH  Google Scholar 

  6. C. D. Maranas, C. A. Floudas, and P. M. Pardalos, “New results in the packing of equal circles in a square,” Discrete Math., 142, 287–293 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  7. M. Locatelli and U. Raber, “Packing equal circles in a square: A deterministic global optimization approach,” Discrete Appl. Math., 122, 139–166 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  8. P. G. Szaro, M. C. Markot, and T. Csendes, “Global optimization in geometry - circle packing into the square,” in: C. Audel, P. Hancen, and G. Savard (eds.), Essays and Surveys in Global Optimization, Kluwer, Dordrecht (2005), pp. 233–266.

    Google Scholar 

  9. Mhand Hifi and Rym M’Hallah, “A literature review on circle and sphere packing problems: Models and methodologies,” Adv. Oper. Res. (2009) (doi:10.1155/2009/150624).

    Google Scholar 

  10. K. Anstreicher, “On convex relaxations for quadratically constrained quadratic programming,” Math. Program., No. 136, 233–251 (2012).

  11. L. Vanderberghe and S. Boyd, “Semidefinite programming,” SIAM Rev., No. 38, 49–95 (1996).

  12. H. D. Sherali and W. P. Adams, A Reformulation–Linearization Technique for Solving Discrete and Continuous Nonconvex Problems, Kluwer, Dordrecht (1998).

    Google Scholar 

  13. Y. Yajima and T. Fujie, “A polyhedral approach for nonconvex quadratic programming problems with box constraints,” J. Global Optim., No. 13, 151–170 (1998).

  14. O. A. Berezovskyi and P. I. Stetsyuk, “An approach to determining Shor’s dual quadratic estimates,” Cybernetics and Systems Analysis. 44, No. 2, 225–234 (2008).

    Article  MathSciNet  Google Scholar 

  15. O. A. Berezovskyi, “On the accuracy of dual bounds for quadratic extremum problems,” Cybernetics and Systems Analysis, 48, No. 1, 26–30 (2012).

    Article  MATH  Google Scholar 

  16. T. Fujit and M. Kojima, “Semidefinite programming relaxation for nonconvex quadratic problems,” J. Global Optim., No. 10, 367–380 (1997).

  17. N. Z. Shor and O. A. Berezovskyi, “New algorithms for the weighted maximum cut problem on graphs,” Cybernetics and Systems Analysis, 31, No. 2, 240–245 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  18. B. N. Pshenichnyi, Linearization Method [in Russian], Nauka, Moscow (1983).

    Google Scholar 

  19. K. Anstreicher, “Semidefinite programming versus the reformulation–linearization technique for nonconvex problems,” J. Global Optim., No. 43, 471–484 (2009).

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to O. A. Berezovskyi.

Additional information

Translated from Kibernetika i Sistemnyi Analiz, No. 4, pp. 170–179, July–August, 2014.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Berezovskyi, O.A. On the Problem of Packing Spheres in a Cube. Cybern Syst Anal 50, 634–642 (2014). https://doi.org/10.1007/s10559-014-9652-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10559-014-9652-2

Keywords

Navigation