Second-Order Cone Relaxations for Binary Quadratic Polynomial Programs

Several types of relaxations for binary quadratic polynomial programs can be obtained using linear, secondorder cone, or semidefinite techniques. In this paper, we propose a general framework to construct conic relaxations for binary quadratic polynomial programs based on polynomial programming. Using our framework, we re-derive previous relaxation schemes and provide new ones. In particular, we present three relaxations for binary quadratic polynomial programs. The first two relaxations, based on second-order cone and semidefinite programming, represent a significant 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, semidefinite-based relaxations for binary quadratic polynomial problems can be used only to solve small to mid-size instances. To improve the computational efficiency for solving such problems, we propose a third relaxation based purely on second-order cone programming. Computational tests on different classes of non-convex binary quadratic polynomial problems, including quadratic knapsack problems, show that the second-order cone-based relaxation outperforms the semidefinite-based relaxations that are proposed in the literature in terms of computational efficiency and is comparable in terms of bounds.