Large-Scale Robust Beamforming via $\ell _{\infty }$ -Minimization

Abstract

In this paper, linearly constrained and robust $\ell _{\infty }$ -norm beamforming techniques are proposed for non-Gaussian signals. A conventional approach for $\ell _{\infty }$ -minimization needs to solve a linear programming (LP) or second-order cone programming (SOCP). However, this strategy is computationally prohibitive for “big data” because the existing algorithms for LP or SOCP, such as simplex method or interior point method, can only solve small- or medium-scale problems. In this paper, the alternating direction method of multipliers (ADMM) is devised for large-scale $\ell _{\infty }$ -beamforming problems, where the core subproblems can be formulated concisely as a linearly or second-order cone constrained least squares and the proximity operator of the $\ell _{\infty }$ -norm in each iteration. Remarkably, a linear-time complexity algorithm is devised that efficiently computes the $\ell _{\infty }$ -norm proximity operator. Simulation results verify the high efficiency of the ADMM and the superiority of the $\ell _{\infty }$ -norm beamforming techniques over several representative beamformers, indicating that its performance can approach the optimal upper bound.