Downlink beamforming and power control with per-antenna power constraints

We study a system where a base station (BS) with an antenna array serves multiple users. Each user has a single antenna. Due to practical limitations, each antenna of the BS has an individual power constraint. Assume that only long-term channel state information (CSI) is available. The objective is to minimize the downlink transmission power under the users´ signal-to-interference-plus-noise ratio (SINR) constraints and the per-antenna power constraints. Mathematically, the objective is to find the optimal solution of a non-convex optimization problem. With dual decomposition, the problem is decomposed into two levels. For the lower level problem, we propose a novel iterative method based on the theory of matrix pencils. For the higher level problem, we propose a subgradient method and prove its convergence. Assembling all the computation methods, we obtain a beamforming and power control algorithm. At last, numerical results are provided to illustrate the performance of the algorithm.