  In this paper, we propose a multi-input multi-output transmit beamforming
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, subject to the transmit power budget. Obtaining the global solution for
the formulated nonconvex problem is a challenging task, since the sum-rate
maximization problem itself (even without considering the sensing metric) is
known to be NP-hard. The main contributions of this paper are threefold.
Firstly, we derive an optimal closed-form solution to the formulated problem in
the single-user case and the multi-user case where the channel vectors of
different users are orthogonal. Secondly, for the general multi-user case, we
propose a novel branch and bound (B\&B) algorithm based on the McCormick
envelope relaxation. The proposed algorithm is guaranteed to find the globally
optimal solution to the formulated problem. Thirdly, we design a graph neural
network (GNN) based pruning policy to determine irrelevant nodes that can be
directly pruned in the proposed B\&B algorithm, thereby significantly reducing
the number of unnecessary enumerations therein and improving its computational
efficiency. Simulation results show the efficiency of the proposed vanilla and
GNN-based accelerated B\&B algorithms.
