Joint Beamforming Optimization for Integrated Sensing and Communication in Cell-Free Massive MIMO Systems

Joint beamforming optimization is a fundamental challenge in achieving efficient integrated sensing and communication (ISAC) for cell-free massive multiple-input multiple-output (MIMO) systems. In the beamforming-vector domain, the design problem is nonconvex because both the transmit-side sensing objective and the user signal-to-interference-plus-noise ratio (SINR) constraints depend quadratically on the transmit beamforming vectors. A common approach is semidefinite relaxation (SDR), where the rank-one constraints on the transmit covariance matrices are removed to obtain a convex semidefinite programming (SDP) problem. However, the relaxed SDP solution may have rank greater than one and therefore cannot be directly mapped to feasible transmit beamforming vectors. Consequently, Gaussian randomization is required to recover rank-one beamforming candidates. In contrast to the SDR approach, this paper proposes a Difference-of-Convex Algorithm (DCA)-based optimization framework that directly enforces the rank-one structure during the optimization process. Specifically, the rank-one requirement is represented by a DC penalty term on each transmit covariance matrix, which regularizes the iterates toward rank-one beamforming covariance matrices. Based on this framework, we develop a DCA implementation tailored to coordinated cell-free architectures, incorporating efficient subgradient computation and an adaptive penalty scheduling mechanism. Simulation results show that the proposed mechanism achieves similar sensing SNR and minimum communication SINR to the Joint Sensing and Communication optimization based on the Semidefinite Programming (JSC-SDP) mechanism. Furthermore, the proposed DCA reduces the rank-1 violation rate by up to 81% and reduces the normalized communication-sensing beam overlap of approximately 33.3%.