Robust Adaptive Beamforming for General-Rank Signal Model With Positive Semi-Definite Constraint via POTDC

The robust adaptive beamforming (RAB) problem for general-rank signal model with an additional positive semi-definite constraint is considered. Using the principle of the worst-case performance optimization, such RAB problem leads to a difference-of-convex functions (DC) optimization problem. The existing approaches for solving the resulted non-convex DC problem are based on approximations and find only suboptimal solutions. Here, we aim at finding the globally optimal solution for the non-convex DC problem and clarify the conditions under which the solution is guaranteed to be globally optimal. Particularly, we rewrite the problem as the minimization of a one-dimensional optimal value function (OVF). Then, the OVF is replaced with another equivalent one, for which the corresponding optimization problem is convex. The new one-dimensional OVF is minimized iteratively via polynomial time DC (POTDC) algorithm. We show that the POTDC converges to a point that satisfies Karush-Kuhn-Tucker (KKT) optimality conditions, and such point is the global optimum under certain conditions. Towards this conclusion, we prove that the proposed algorithm finds the globally optimal solution if the presumed norm of the mismatch matrix that corresponds to the desired signal covariance matrix is sufficiently small. The new RAB method shows superior performance compared to the other state-of-the-art general-rank RAB methods.