Skip to main content
SpringerLink
Log in

D-optimal matrices via quadratic integer optimization

  • Published:
Journal of Heuristics Aims and scope Submit manuscript

Abstract

We show how to express the problem of searching for D-optimal matrices as a Linear and Quadratic Integer Optimization problem. We also focus our attention in the case where the size of the circulant submatrices that are used to construct a D-optimal matrix is a multiple of 3. In this particular case, we describe some additional combinatorial and number-theoretic characteristics that a solution of the D-optimal matrix problem must possess. We give some solutions for some quite challenging D-optimal matrix problems that can be used as benchmarks to test the efficiency of Linear and Quadratic Integer Optimization algorithms.

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

  • Babapour, A.A., Seifi, A.: Optimal design of multi-response experiments using semi-definite programming. Optim. Eng. 10(1), 75–90 (2009)

    Article  MathSciNet  Google Scholar 

  • Cohn, J.H.E.: A D-optimal design of order 102. Discrete Math. 102, 61–65 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  • Djokovic, D.Z.: On maximal (1,−1)-matrices of order 2n,n odd. Rad. Mat. 7(2), 371–378 (1991)

    MathSciNet  MATH  Google Scholar 

  • Djokovic, D.Z.: Some new D-optimal designs. Australas. J. Comb. 15, 221–231 (1997)

    MATH  Google Scholar 

  • Floudas, C.A., Pardalos, P.M. (eds.): Encyclopedia of Optimization, vols. I–VI. Kluwer Academic, Dordrecht (2001)

    Google Scholar 

  • Gysin, M.: New D-optimal designs via cyclotomy and generalised cyclotomy. Australas. J. Comb. 15, 247–255 (1997)

    MathSciNet  MATH  Google Scholar 

  • Gysin, M., Seberry, J.: An experimental search and new combinatorial designs via a generalisation of cyclotomy. J. Comb. Math. Comb. Comput. 27, 143–160 (1998)

    MathSciNet  MATH  Google Scholar 

  • Kharaghani, H., Orrick, W.: D-optimal matrices. In: Colbourn, C.J., Dinitz, J.H. (eds.) Handbook of Combinatorial Designs, 2nd edn. Discrete Mathematics and Its Applications. Chapman & Hall/CRC, Boca Raton (2007)

    Google Scholar 

  • Kotsireas, I.S., Koukouvinos, C.: A computational algebraic approach for saturated D-optimal designs with \(n\equiv2\pmod{4}\) observations. Util. Math. 71, 197–207 (2006)

    MathSciNet  MATH  Google Scholar 

  • Kotsireas, I.S., Koukouvinos, C., Seberry, J.: Weighing matrices and string sorting. Ann. Comb. 13(3), 305–313 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Kotsireas, I.S., Koukouvinos, C., Pardalos, P.M., Shylo, O.V.: Periodic complementary binary sequences and linear and quadratic integer optimization algorithms. J. Comb. Optim. 20, 63–75 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Wolkowicz, H., Saigal, R., Vandenberghe, L. (eds.): Handbook of Semidefinite Programming. Theory, Algorithms, and Applications. International Series in Operations Research & Management Science, vol. 27. Kluwer Academic, Boston (2000)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Phys. & Comp. Science, Wilfrid Laurier University, Waterloo, ON, N2L 3C5, Canada

    I. S. Kotsireas

  2. Industrial and Systems Engineering Department, Center for Applied Optimization, Gainesville, FL, 32611-6595, USA

    P. M. Pardalos

Authors
  1. I. S. Kotsireas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. M. Pardalos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to I. S. Kotsireas.

Additional information

This research is partially supported by NSF, AirForce and NSERC grants.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kotsireas, I.S., Pardalos, P.M. D-optimal matrices via quadratic integer optimization. J Heuristics 19, 617–627 (2013). https://doi.org/10.1007/s10732-011-9173-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10732-011-9173-3

Keywords

Search

Navigation