Acyclic semidefinite approximations of quadratically constrained quadratic programs

Quadratically constrained quadratic programs (QCQPs) belong to a class of nonconvex optimization problems that are NP-hard in general. Recent results have shown that QCQPs having acyclic graph structure can be solved in polynomial time, provided that their constraints satisfy a certain technical condition. In this paper, we consider complex QCQPs with arbitrary graph structure and investigate the extent to which it is possible to apply structured perturbations on the problem data to yield acyclic QCQPs having optimal solutions satisfying certain approximation guarantees. Specifically, we provide sufficient conditions under which the perturbed QCQP can be solved in polynomial time to yield a feasible solution to the original QCQP and derive an explicit bound on the performance of said solution in the worst case.