Mathematical Programming

, Volume 136, Issue 2, pp 301–323 | Cite as

The quadratic Graver cone, quadratic integer minimization, and extensions

  • Jon Lee
  • Shmuel Onn
  • Lyubov Romanchuk
  • Robert Weismantel
Full Length Paper Series B


It has been shown in a number of recent papers that Graver bases methods enable to solve linear and nonlinear integer programming problems in variable dimension in polynomial time, resulting in a variety of applications in operations research and statistics. In this article we continue this line of investigation and show that Graver bases also enable to minimize quadratic and higher degree polynomial functions which lie in suitable cones. These cones always include all separable convex polynomials and typically more.


Integer programming Discrete optimization Graver basis Quadratic optimization Semidefinite programming Polynomial optimization 

Mathematics Subject Classification

52B 52C 62H 68Q 68R 90B 90C 



We thank the referees for helpful feedback which improved the presentation of this article.


  1. 1.
    Berstein, Y., Onn, S.: The Graver complexity of integer programming. Ann. Comb. 13, 289–296 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    De Loera, J., Hemmecke, R., Onn, S., Rothblum, U.G., Weismantel, R.: Convex integer maximization via Graver bases. J. Pure Appl. Algebra 213, 1569–1577 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    De Loera, J., Hemmecke, R., Onn, S., Weismantel, R.: N-fold integer programming. Discret. Optim. 5, 231–241 (2008) (Volume in memory of George B. Dantzig)Google Scholar
  4. 4.
    De Loera, J., Onn, S.: All linear and integer programs are slim 3-way transportation programs. SIAM J. Optim. 17, 806–821 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Dobra, A., Fienberg, S.E., Rinaldo, A., Slavković, A., Zhou, Y.: Algebraic statistics and contingency table problems: log-linear models, likelihood estimation, and disclosure limitation. In: Emerging Applications of Algebraic Geometry: IMA Volumes in Mathematics and its Applications, vol. 148, pp. 63–88. Springer, Berlin (2009)Google Scholar
  6. 6.
    Gordan, P.: Über die Auflösung linearer Gleichungen mit reellen Coefficienten. Math. Ann. 6, 23–28 (1873)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Graver, J.E.: On the foundation of linear and integer programming I. Math. Program. 9, 207–226 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Hemmecke, R., Onn, S., Weismantel, R.: A polynomial oracle-time algorithm for convex integer minimization. Math. Program. 126, 97–117 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Hemmecke, R., Onn, S., Weismantel, R.: N-fold integer programming and nonlinear multi-transshipment. Optim. Lett. 5, 13–25 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Motzkin, T.S.: The multi-index transportation problem. Bull. Am. Math. Soc. 58, 494 (1952)Google Scholar
  11. 11.
    Onn, S.: Nonlinear Discrete Optimization. Zurich Lectures in Advanced Mathematics. European Mathematical Society, Zurich (2010)CrossRefGoogle Scholar
  12. 12.
    Schrijver, A.: Theory of Linear and Integer Programming. Wiley, New York (1986)zbMATHGoogle Scholar
  13. 13.
    Sebö, A.: Hilbert bases, Carathéodory’s theorem and combinatorial optimization. In: Proc. IPCO 1–1st Conference on Integer Programming and Combinatorial Optimization, pp. 431–455. University of Waterloo Press, Waterloo (1990)Google Scholar
  14. 14.
    Santos, F., Sturmfels, B.: Higher Lawrence configurations. J. Comb. Theory Ser. A 103, 151–164 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Van Der Waerden, B.L.: Algebra. Frederick Ungar Publishing, New York (1970)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2012

Authors and Affiliations

  • Jon Lee
    • 1
  • Shmuel Onn
    • 2
  • Lyubov Romanchuk
    • 2
  • Robert Weismantel
    • 3
  1. 1.IOE DepartmentUniversity of MichiganAnn ArborUSA
  2. 2.Technion—Israel Institute of TechnologyHaifaIsrael
  3. 3.ETH ZürichZürichSwitzerland

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