Globally Optimal Beamforming Design for Integrated Sensing and Communication Systems

In this paper, we propose a multi-input multi-output (MIMO) beamforming transmit optimization model for joint radar sensing and multi-user communications, where the design of the beamformers is formulated as an optimization problem whose objective is a weighted combination of the sum rate and the Cram\'{e}r-Rao bound (CRB), subject to the transmit power budget constraint. The formulated problem is challenging to obtain a global solution, because the sum rate maximization (SRM) problem itself (even without considering the sensing metric) is known to be NP-hard. In this paper, we propose an efficient global branch-and-bound algorithm for solving the formulated problem based on the McCormick envelope relaxation and the semidefinite relaxation (SDR) technique. The proposed algorithm is guaranteed to find the global solution for the considered problem, and thus serves as an important benchmark for performance evaluation of the existing local or suboptimal algorithms for solving the same problem.