Design of Robust Polynomial Beamformers Using Worst Case Performance Optimization via Alternating Direction Method of Multipliers

This article studies the design of robust polynomial beamformers (RPBs) for microphone arrays, which enable dynamic beam steering via simple online parameter tuning, under the criterion of worst case performance optimization (WCPO). Although the WCPO criterion has been widely adopted in the design of various robust beamformers, conventional approaches using the WCPO criterion may not be applicable to the RPB design due to the conservativeness problem resulting from the special structure of polynomial beamformers. In this article, we propose to design the RPBs based on semidefinite programming (SDP) using the WCPO criterion. Nevertheless, the resulting design optimization problems are highly computationally demanding. To overcome the computational difficulty, we then propose to solve the high-dimensional optimization problems for the RPB design using the alternating direction method of multipliers (ADMM) algorithm. By introducing auxiliary variables in the framework of ADMM, the original high-dimensional optimization problems are decomposed into several subproblems that are easier to handle. The algorithms to solve the corresponding subproblems are developed. Simulation and real-world experimental results show that the proposed SDP-based RPB design performs much better than its existing counterpart. Moreover, the computational efficiency can be significantly improved by using the proposed ADMM algorithm, while the existing solution is very time-consuming and even fails.