New bounds on the unconstrained quadratic integer programming problem

  • Authors:

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

  • Affiliations:

  • School of Engineering and Mathematical Sciences, City University, London, UK EC1V 0HB;Control and Power Group, Department of Electrical and Electronic Engineering, Imperial College, London, UK SW7-2BT;Department of Electrical and Electronic Engineering, Imperial College, London, UK SW7-2BT;Department of Electrical and Electronic Engineering, Imperial College, London, UK SW7-2BT

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2007

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Abstract

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 驴 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.