Journal of Heuristics

, Volume 19, Issue 4, pp 617–627 | Cite as

D-optimal matrices via quadratic integer optimization

  • I. S. Kotsireas
  • P. M. Pardalos


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.


Periodic autocorrelation function Linear and quadratic integer optimization Algorithms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Babapour, A.A., Seifi, A.: Optimal design of multi-response experiments using semi-definite programming. Optim. Eng. 10(1), 75–90 (2009) MathSciNetCrossRefGoogle Scholar
  2. Cohn, J.H.E.: A D-optimal design of order 102. Discrete Math. 102, 61–65 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  3. Djokovic, D.Z.: On maximal (1,−1)-matrices of order 2n,n odd. Rad. Mat. 7(2), 371–378 (1991) MathSciNetzbMATHGoogle Scholar
  4. Djokovic, D.Z.: Some new D-optimal designs. Australas. J. Comb. 15, 221–231 (1997) zbMATHGoogle Scholar
  5. Floudas, C.A., Pardalos, P.M. (eds.): Encyclopedia of Optimization, vols. I–VI. Kluwer Academic, Dordrecht (2001) Google Scholar
  6. Gysin, M.: New D-optimal designs via cyclotomy and generalised cyclotomy. Australas. J. Comb. 15, 247–255 (1997) MathSciNetzbMATHGoogle Scholar
  7. 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) MathSciNetzbMATHGoogle Scholar
  8. 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
  9. 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) MathSciNetzbMATHGoogle Scholar
  10. Kotsireas, I.S., Koukouvinos, C., Seberry, J.: Weighing matrices and string sorting. Ann. Comb. 13(3), 305–313 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  11. 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) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 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

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Phys. & Comp. ScienceWilfrid Laurier UniversityWaterlooCanada
  2. 2.Industrial and Systems Engineering DepartmentCenter for Applied OptimizationGainesvilleUSA

Personalised recommendations