An Optimality Gap Test for a Semidefinite Relaxation of a Quadratic Program with Two Quadratic Constraints

We propose a necessary and sufficient test to determine whether solution for general quadratic program with two constraints (QC2QP) can be computed from that of specific convex semidefinite relaxation, in which case we say there is no optimality gap. Originally intended solve nonconvex optimal control problem, consider the cost both QC2QP may nonconvex. obtained our test, also ascertains when strong duality holds, by generalizing closely-related method Ai Zhang. An extension was because, while proposed Zhang allows constraints, it requires at least one strictly convex. In order illustrate usefulness applied examples do not satisfy assumptions required prior methods. Our guarantees gap first example -- relaxation used establish an exists second.