This paper addresses the weighted sum-rate (WSR) maximization problem in a downlink distributed antenna system subject to per-cluster power constraints. This optimization scenario presents significant challenges due to the high dimensionality of beamforming variables in dense antenna deployments and the structural complexity of multiple independent power constraints. To overcome these difficulties, we generalize the low-dimensional subspace property--previously established for sum-power constraints--to the per-cluster power constraint case. We prove that all stationary-point beamformers reside in a reduced subspace spanned by the channel vectors of the corresponding antenna cluster. Leveraging this property, we reformulate the original high-dimensional constrained problem into an unconstrained optimization task over a product of ellipsoidal manifolds, thereby achieving significant dimensionality reduction. We systematically derive the necessary Riemannian geometric structures for this specific manifold, including the tangent space, Riemannian metric, orthogonal projection, retraction, and vector transport. Subsequently, we develop a tailored Riemannian conjugate gradient algorithm to solve the reformulated problem. Numerical simulations demonstrate that the proposed algorithm achieves the same local optima as standard benchmarks, such as the weighted minimum mean square error (WMMSE) method and conventional manifold optimization, but with substantially higher computational efficiency and scalability, particularly as the number of antenna clusters increases.

翻译：本文研究了下行分布式天线系统中在每簇功率约束下的加权和速率（WSR）最大化问题。该优化场景因密集天线部署中波束成形变量的高维度以及多个独立功率约束的结构复杂性而面临显著挑战。为克服这些困难，我们将此前针对总功率约束建立的低维子空间性质推广至每簇功率约束情形。我们证明所有驻点波束成形器均位于对应天线簇信道向量张成的约化子空间中。基于该性质，我们将原始高维约束问题重构为椭球流形乘积上的无约束优化任务，从而实现了显著的维度约减。我们系统推导了该特定流形所需的黎曼几何结构，包括切空间、黎曼度量、正交投影、回缩和向量传输。随后，我们开发了定制化的黎曼共轭梯度算法以求解重构后的问题。数值仿真表明，所提算法能够达到与加权最小均方误差（WMMSE）方法及传统流形优化等标准基准相同的局部最优解，但计算效率与可扩展性显著提升，尤其当天线簇数量增加时。