We study the exactness of the semidefinite programming (SDP) relaxation of quadratically constrained quadratic programs (QCQPs). With the aggregate sparsity matrix from the data matrices of a QCQP with n variables, the rank and positive semidefiniteness of the matrix are examined. We prove that if the rank of the aggregate sparsity matrix is not less than \(n-1\) and the matrix remains positive semidefinite after replacing some off-diagonal nonzero elements with zeros, then the standard SDP relaxation provides an exact optimal solution for the QCQP under feasibility assumptions. In particular, we demonstrate that QCQPs with forest-structured aggregate sparsity matrix, such as the tridiagonal or arrow-type matrix, satisfy the exactness condition on the rank. The exactness is attained by considering the feasibility of the dual SDP relaxation, the strong duality of SDPs, and a sequence of QCQPs with perturbed objective functions, under the assumption that the feasible region is compact. We generalize our result for a wider class of QCQPs by applying simultaneous tridiagonalization on the data matrices. Moreover, simultaneous tridiagonalization is applied to a matrix pencil so that QCQPs with two constraints can be solved exactly by the SDP relaxation.

我们研究了二次约束二次规划 (QCQP) 的半定规划 (SDP) 松弛的准确性。使用来自具有n 个变量的 QCQP 的数据矩阵的聚合稀疏矩阵，检查矩阵的秩和正半定性。我们证明如果聚合稀疏矩阵的秩不小于\(n-1\)并且矩阵在用零替换一些非对角非零元素后仍然是半正定的，那么标准 SDP 松弛在可行性假设下为 QCQP 提供了精确的最优解。特别是，我们证明了具有森林结构聚合稀疏矩阵（例如三对角矩阵或箭头型矩阵）的 QCQP 满足秩的精确性条件。在可行区域紧凑的假设下，通过考虑双重 SDP 松弛的可行性、SDP 的强对偶性以及具有扰动目标函数的 QCQP 序列来获得准确性。我们通过对数据矩阵应用同步三对角化，将我们的结果推广到更广泛的 QCQP 类别。而且，