We consider a non-convex quadratic program (QP) with linear and convex quadratic constraints that arises from a broad range of applications and is known to be NP-hard. In this paper, we first prove that the alternative direction method converges to a local solution of the underlying QP problem. We then propose a new branch-and-cut algorithm that finds a globally optimal solution to the underlying QP problem within a pre-specified 𝜖-tolerance by integrating the alternative direction method with semidefinite relaxation and disjunctive cut techniques. We establish the global convergence of the algorithm and estimate its complexity. Preliminary numerical results demonstrate that the proposed algorithm can effectively find a globally optimal solution to medium-scale QP instances in which the number of negative eigenvalues of the Hessian matrix in the objective function is less than or equals 20.

我们考虑具有线性和凸二次约束的非凸二次规划 (QP)，该约束源于广泛的应用，并且已知为 NP-hard。 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 在本文中，我们首先证明了替代方向方法收敛于底层 QP 问题的局部解。然后，我们提出了一种新的分支切割算法，该算法可以在预先指定的𝜖内找到底层 QP 问题的全局最优解-通过将替代方向方法与半定松弛和析取切割技术相结合来提高容差。我们建立算法的全局收敛性并估计其复杂度。初步数值结果表明，该算法可以有效地找到目标函数中Hessian矩阵的负特征值个数小于等于20的中等规模QP实例的全局最优解。