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 solve the non-convex DC problem rigorously and give arguments suggesting that the solution is globally optimal. Particularly, we rewrite the problem as the minimization of a one-dimensional optimal value function whose corresponding optimization problem is non-convex. Then, the optimal value function is replaced with another equivalent one, for which the corresponding optimization problem is convex. The new one-dimensional optimal value function is minimized iteratively via polynomial time DC (POTDC) algorithm.We show that our solution satisfies the Karush-Kuhn-Tucker (KKT) optimality conditions and there is a strong evidence that such solution is also globally optimal. Towards this conclusion, we conjecture that the new optimal value function is a convex function. The new RAB method shows superior performance compared to the other state-of-the-art general-rank RAB methods.