Journal of Global Optimization

, Volume 39, Issue 4, pp 543–554 | Cite as

New bounds on the unconstrained quadratic integer programming problem

  • G. D. Halikias
  • I. M. Jaimoukha
  • U. Malik
  • S. K. Gungah
Original Paper


We consider the maximization \(\gamma = \max\{x^{T}\!Ax : x\in \{-1, 1\}^n\}\) for a given symmetric \(A\in\mathcal{R}^{n\times n}\). It was shown recently, using properties of zonotopes and hyperplane arrangements, that in the special case that A has a small rank, so that A = VV T where \(V\in\mathcal{R}^{n\times m}\) with m < n, then there exists a polynomial time algorithm (polynomial in n for a given m) to solve the problem \(\max\{x^TV V^Tx : x\in \{-1, 1\}^n\}\). In this paper, we use this result, as well as a spectral decomposition of A to obtain a sequence of non-increasing upper bounds on γ with no constraints on the rank of A. We also give easily computable necessary and sufficient conditions for the absence of a gap between the solution and its upper bound. Finally, we incorporate the semidefinite relaxation upper bound into our scheme and give an illustrative example.


Quadratic integer programming Semidefinite relaxation Zonotope Hyperplane arrangements 


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Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • G. D. Halikias
    • 1
  • I. M. Jaimoukha
    • 2
  • U. Malik
    • 3
  • S. K. Gungah
    • 3
  1. 1.School of Engineering and Mathematical SciencesCity UniversityLondonUK
  2. 2.Control and Power Group, Department of Electrical and Electronic EngineeringImperial CollegeLondonUK
  3. 3.Department of Electrical and Electronic EngineeringImperial CollegeLondonUK

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