In this paper, we propose a novel low-dimensional semidefinite programming (SDP) relaxation for convex quadratically constrained nonconvex quadratic programming problems. This new relaxation is derived by simultaneous matrix diagonalization method under the difference of convex decomposition scheme. The highlight of the relaxation is the low dimensionality of the positive semidefinite constraint, which only depends on the number of negative eigenvalues in the objective function. We prove that a mixed SOCP and SDP relaxation appeared in the literature is equivalent to the proposed relaxation, while the proposed relaxation has fewer constraints. We also provide conditions under which the proposed relaxation is as tight as the classical SDP relaxation and provides an optimal value for the original problem. For general cases, a spatial branch-and-bound algorithm is designed for finding a global optimal solution. Extensive numerical experiments support that the proposed algorithm outperforms two cutting-edge algorithms when the problem size is medium or large and the number of negative eigenvalues in the nonconvex objective function is relatively small.

中文翻译：

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一种非凸二次规划的基于低维SDP松弛的空间分支定界方法

在本文中，我们针对凸二次约束非凸二次规划问题提出了一种新颖的低维半定规划（SDP）松弛。这种新的松弛是在凸分解方案的差异下通过同时矩阵对角化方法得出的。松弛的亮点是正半定约束的低维性，它仅取决于目标函数中负特征值的数量。我们证明，文献中出现的混合SOCP和SDP松弛与所提​​出的松弛等效，而所提出的松弛具有较少的约束。我们还提供了条件，在该条件下，拟议的松弛度与经典SDP松弛度一样严格，并且为原始问题提供了最佳值。对于一般情况，设计了一种空间分支定界算法来寻找全局最优解。大量的数值实验证明，当问题的大小为中等或较大且非凸目标函数中的负特征值的数量较小时，该算法的性能要优于两个前沿算法。