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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

Abstract

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.

Keywords

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

Mathematics Subject Classification

52B 52C 62H 68Q 68R 90B 90C 

Notes

Acknowledgments

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

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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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