On the tightness of an SDP relaxation for homogeneous QCQP with three real or four complex homogeneous constraints

In this paper, we consider the problem of minimizing a general homogeneous quadratic function, subject to three real or four complex homogeneous quadratic inequality or equality constraints. For this problem, we present a sufficient and necessary test condition to detect whether its standard semi-definite programming (SDP) relaxation is tight or not. This test condition is based on only an optimal solution pair of the SDP relaxation and its dual. When the tightness is confirmed, a global optimal solution of the original problem is found simultaneously in polynomial-time. While the tightness does not hold, the SDP relaxation and its dual are proved to have the unique optimal solutions. Moreover, the Lagrangian version of such the test condition is specified for non-homogeneous cases. Based on the Lagrangian version, it is proved that several latest sufficient conditions to test the SDP tightness are contained by our test condition under the situation of two constraints. Thirdly, as an application of the test condition, S-lemma and Yuan’s lemma are generalized to three real and four complex quadratic forms first under certain exact conditions, which improves some classical results in literature. Finally, a counterexample is presented to show that the test condition cannot be simply extended to four real or five complex homogeneous quadratic constraints.