Robust Adaptive Beamforming with Nonconvex Union of Multiple Steering Vector Uncertainty Sets

A robust adaptive beamforming (RAB) problem maximizing the worst-case signal-to-interference-plus-noise ratio (SINR) over the union of small uncertainty sets, each with a similarity constraint on the desired signal steering vector, is formulated and recast into a minimization problem with a convex quadratic objective function under constraints in the form of difference of convex quadratic functions. It has been proved in the literature that the union of the small uncertainty sets can be utilized to sufficiently enlarge the overall uncertainty set that a single large sphere set cannot cover. A sequential second-order cone programming (SOCP) approximation algorithm is proposed in this paper, which has lower computational cost but improved beamformer output SINR in comparison with existing algorithms. It is also verified that the sequence of the optimal values of the SOCPs is nonincreasing and bounded, and the optimal solutions of the SOCPs are always feasible for the quadratic problem, and the sequence of the solutions converges to a locally optimal solution for the RAB problem. In addition, every small uncertainty set is generalized by adding one more fixed norm constraint, which is practical since the norm of the signal steering vector is often equal to the square root of the number of the array antennas. Consequently, the worst-case SINR maximization problem with the union of the generalized small uncertainty sets is converted into a quadratic matrix inequality (QMI) problem via the strong duality of semidefinite programming, and a rank-reduction solution procedure is employed to find a rank-one optimal solution for the linear matrix inequality relaxation problem of the QMI problem. Moreover, another algorithm is proposed for an MVDR RAB problem with the nonconvex union of the extended small uncertainty sets, and an optimal MVDR RAB beamformer is obtained by separately solving the corresponding homogeneous quadratically constrained quadratic programming (QCQP) subproblems, and selecting the best solution from the set of all the globally optimal solutions for the QCQP subproblems to yield the maximal array output power. Simulation examples are presented to demonstrate the improved performance of the proposed beamformers in terms of the array output SINR, running time and number of iterations, as well as the normalized beampattern, compared with existing beamformers.