Integrated sensing and communication (ISAC) has been envisioned as a key enabler in the next generation wireless networks. In this paper, we consider the joint information and sensing beamforming design in a multi-user and multi-target multi-input multi-output (MUMT-MIMO) ISAC system, where a transmit BS and a sensing BS collaborate to sense targets, and the transmit BS sends information streams to communication users at the same time. To optimize the sensing performance and guarantee the communication throughput, we formulate a joint beamforming design problem to minimize the trace of the weighted Cramer-Rao bound (CRB) of target parameters subject to the sum rate constraint. The problem is challenging to solve due to the intricate non-convex objective function and constraints. We firstly exploit the weighted mean square error minimization (WMMSE) and semidefinite relaxation (SDR) techniques to devise a WMMSE-SDR algorithm that can achieve a KKT point of the problem.
The SDR can be shown to be tight for a sub-problem in the WMMSE-SDR algorithm, which implies zero duality for the sub-problem. Based on this property and fractional programming (FP) techniques, we further reformulate the beamforming problem as a min-max form with simple constraints which then can be efficiently solved by first-order min-max optimization algorithms. Finally, the proposed algorithms are evaluated extensively in simulation. Numerical results show that both proposed algorithms can achieve promising performance in sensing and communication, and the low-complexity algorithm has a significantly reduced computation time.