Exact SDP relaxations of quadratically constrained quadratic programs with forest structures

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