Doubly constrained robust capon beamformer with ellipsoidal uncertainty sets

The doubly constrained robust (DCR) Capon beamformer with a spherical uncertainty set was introduced and studied by Stoica and Wang. Here, we consider the generalized DCR problem (GDCR) in which the uncertainty set is an ellipsoid rather than a sphere. Although, as noted previously by Stoica and Wang, this problem is nonconvex and appears to be intractable, we show that it can be solved efficiently. In fact, we prove that the GDCR beamformer can be obtained as a solution to a convex optimization problem. To this end, we first derive a strong duality result for nonconvex quadratic programs with two quadratic constraints over the complex domain. Specializing the results to our context leads to a semidefinite programming formulation of the GDCR beamformer.