Second-Order Cone Relaxations for Binary Quadratic Polynomial Programs

SIAM Journal on Optimization (Impact Factor: 1.83). 01/2011; 21(1):391-414. DOI: 10.1137/100802190
Source: DBLP


techniques. In this paper, we propose a general framework to construct conic relaxations for binary quadratic polynomial programs based on polynomial programming. Using our frame- work, we re-derive previous relaxation schemes and provide new ones. In particular, we present three relax- ations for binary quadratic polynomial programs. The rst two relaxations, based on second-order cone and semidenite programming, represent a signicant improvement over previous practical relaxations for several classes of non-convex binary quadratic polynomial problems. From a practical point of view, due to the computational cost, semidenite-based relaxations for binary quadratic polynomial problems can be used only to solve small to mid-size instances. To improve the computational eciency for solving such problems, we propose a third relaxation based purely on second-order cone programming. Computational tests on dif- ferent classes of non-convex binary quadratic polynomial problems, including quadratic knapsack problems, show that the second-order cone-based relaxation outperforms the semidenite-based relaxations that are proposed in the literature in terms of computational eciency and is comparable in terms of bounds.

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Available from: Juan C Vera, Jan 21, 2014
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    • "Problem (7) is one such example that leads to a somewhat concise semi-definite program. We refer the interested reader to Lasserre (2002) and Ghaddar et al. (2011) for a hierarchy of polynomial size semi-definite programming relaxation of mixed-integer quadratic programs for which the integrality gap is known to converge to 1. Based on Proposition 2, it is therefore theoretically possible to find a semi-definite programming model of polynomial size that will generate a solution within a constant factor of the optimal one. Unfortunately, this might often be of little practical relevance. "
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