The Necessary and Sufficient Condition for SDP Relaxation to be Tight in MVDR Estimation

This paper investigates the problem of solving the minimum variance distortionless response (MVDR) robust adaptive beamforming (RAB) by estimating the steering vector of the signal-of-interest (SOI). And this issue can be reformulated as a homogeneous quadratic constrained quadratic programming (HQCQP). We derive the optimal solution of its relaxation by SDP, and subsequently propose methods to figure out its tightness and then achieve a globally optimal solution for HQCQP. It should be noted that a solution with rank one is not guaranteed in existing literatures. Leveraging the recent advancements in SDP, namely, a condition that serves as both a sufficient and necessary test, we verify if the SDP relaxation is tight and then propose effective algorithms to address the problem. Two specific instances of the MVDR have been modeled. Numerical experiments are presented, illustrating the high ratioof the presence of a solution with rank one in the MVDR scenario, and the robust adaptive beamformer with respect to output SINR performs well.