This paper studies the optimal placement of ceiling-mounted metasurfaces (MTSs) to help focus the wireless signal beam onto the target receiver, as inspired by the theatre spotlight. We assume that a total of $M$ MTSs are deployed, and that there are $L$ possible positions for each MTS. The resulting signal-to-noise (SNR) maximization problem is difficult to tackle directly because of the coupling between the placement decisions of the different MTSs. Mathematically, we are faced with a nonlinear discrete optimization problem with $L^M$ possible solutions. A remarkable result shown in this paper is that the above challenging problem can be efficiently solved within $O(ML^2\log(ML))$ time. There are two key steps in developing the proposed algorithm. First, we successfully decouple the placement variables of different MTSs by introducing a continuous auxiliary variable $μ$; the discrete primal variables are now easy to optimize when $μ$ is held fixed, but the optimization problem of $μ$ is nonconvex. Second, we show that the optimization of continuous $μ$ can be recast into a discrete optimization problem with only $LM$ possible solutions, so the optimal $μ$ can now be readily obtained. Numerical results show that the proposed algorithm can not only guarantee a global optimum but also reach the optimal solution efficiently.

翻译：本文研究如何优化部署天花板超表面以将无线信号波束聚焦于目标接收器，其灵感来源于剧场聚光灯。我们假设共部署$M$个超表面，且每个超表面有$L$个可选安装位置。由于不同超表面的部署决策之间存在耦合，由此产生的信噪比最大化问题难以直接求解。从数学角度，我们面临一个具有$L^M$种可能解的非线性离散优化问题。本文证明了一个重要结论：该复杂问题可在$O(ML^2\log(ML))$时间内高效求解。所提算法的开发包含两个关键步骤：首先，通过引入连续辅助变量$μ$成功解耦不同超表面的部署变量——当$μ$固定时，离散原始变量的优化变得简单，但$μ$的优化问题本身是非凸的；其次，我们证明连续变量$μ$的优化可转化为仅含$LM$种可能解的离散优化问题，从而能便捷地获得最优$μ$。数值结果表明，所提算法不仅能保证获得全局最优解，还能高效收敛至最优解。