A standard quadratic program is an optimization problem that consists of minimizing a (nonconvex) quadratic form over the unit simplex. We focus on reformulating a standard quadratic program as a mixed integer linear programming problem. We propose two alternative formulations. Our first formulation is based on casting a standard quadratic program as a linear program with complementarity constraints. We then employ binary variables to linearize the complementarity constraints. For the second formulation, we first derive an overestimating function of the objective function and establish its tightness at any global minimizer. We then linearize the overestimating function using binary variables and obtain our second formulation. For both formulations, we propose a set of valid inequalities. Our extensive computational results illustrate that the proposed mixed integer linear programming reformulations significantly outperform other global solution approaches. On larger instances, we usually observe improvements of several orders of magnitude.

标准二次程序是一个优化问题，包括最小化单位单纯形上的（非凸）二次形式。我们专注于将标准二次程序重新构造为混合整数线性规划问题。我们提出了两种替代方案。我们的第一个公式是基于将标准二次程序转换为具有互补约束的线性程序。然后，我们使用二进制变量来线性化互补性约束。对于第二种公式，我们首先导出目标函数的高估函数，并在任何全局最小化器处建立其紧密度。然后，我们使用二进制变量线性化高估函数并获得第二个公式。对于这两种公式，我们提出了一组有效的不等式。我们广泛的计算结果表明，所提出的混合整数线性规划公式显着优于其他全局求解方法。在较大的情况下，我们通常会观察到几个数量级的改进。