A minimax regret approach to robust beamforming

Minimum variance beamforming, which uses a steering vector that maximizes the signal-to-interference-plus-noise ratio (SINR), is sensitive to estimation error and uncertainty in the steering vector. Robust beamforming attempts to alleviate this sensitivity systematically by incorporating a data uncertainty model in the optimization problem and optimizing an objective over the model; (e.g., maximizing the worst-case SINR, where 'worst' means smallest). However such a worst-case (maximin) approach can be conservative, meaning that its performance loss compared with the optimal beamformer for the true parameters which are unknown can be significant. This dissertation describes a minimax regret approach to robust beamforming with uncertainty model of minimum volume ellipsoid, in which the objective is to minimize the worst-case regret over steering vector mismatch, where 'worst' means largest. This problem can be solved efficiently by an iterative method which uses an alternating sequence of optimization and worst-case analysis steps. Each of the two steps amounts to solving a convex optimization problem. The method typically converges to a solution within 5 iterations. As contributions of this dissertation, the proposed minimax regret beamformer for improved performance can be solved efficiently using convex optimization. Whereas the maximin approach is immoderately conservative, the minimax regret approach is more attractive and less conservative because it benchmarks decision theory with design variables chosen optimally for minimizing the regret of decision. As application, the possibility to replace the worst-case performance maximization problem with the minimax regret problem in for robust optimization signal processing areas is evaluated.