This paper is concerned with the computation of the capacity region of a continuous, Gaussian vector broadcast channel (BC) with covariance matrix constraints. Since the decision variables of the corresponding optimization problem are Gaussian distributed, they can be characterized by a finite number of parameters. Consequently, we develop new Blahut-Arimoto (BA)-type algorithms that can compute the capacity without discretizing the channel. First, by exploiting projection and an approximation of the Lagrange multiplier, which are introduced to handle certain positive semidefinite constraints in the optimization formulation, we develop the Gaussian BA algorithm with projection (GBA-P). Then, we demonstrate that one of the subproblems arising from the alternating updates admits a closed-form solution. Based on this result, we propose the Gaussian BA algorithm with alternating updates (GBA-A) and establish its convergence guarantee. Furthermore, we extend the GBA-P algorithm to compute the capacity region of the Gaussian vector BC with both private and common messages. All the proposed algorithms are parameter-free. Lastly, we present numerical results to demonstrate the effectiveness of the proposed algorithms.

翻译：本文研究具有协方差矩阵约束的连续高斯向量广播信道容量区域的计算问题。由于对应优化问题的决策变量服从高斯分布，它们可通过有限数量的参数进行表征。因此，我们开发了新型Blahut-Arimoto类算法，无需对信道进行离散化即可计算容量。首先，通过利用为处理优化公式中特定半正定约束而引入的投影技术与拉格朗日乘子近似，我们提出了带投影的高斯BA算法。随后，我们证明交替更新过程中产生的子问题存在闭式解。基于此结果，我们提出带交替更新的高斯BA算法并建立其收敛性保证。进一步地，我们将GBA-P算法扩展至同时包含私有消息与公共消息的高斯向量广播信道容量区域计算。所有提出的算法均无需参数调优。最后，我们通过数值实验验证所提算法的有效性。