Optimal Power Aggregation of Reconfigurable Intelligent Surfaces: An Alternating Inner Product Maximization Approach

The reconfigurable intelligent surface (RIS) has garnered considerable attention due to its substantial potential in reconfiguring the electromagnetic environment. In RIS-aided communications, constrained $\ell _{2}$ -norm maximization problems frequently arise due to the phase configuration requirements. This paper investigates a general discrete $\ell _{p}$ -norm maximization problem, with power aggregation through RIS as a specific example. We propose a mathematically concise iterative framework composed of alternating inner product maximizations, which is well-suited for addressing both $\ell _{1}$ - and $\ell _{2}$ -norm maximizations under either discrete or continuous uni-modular variable constraints. The iteration process is proven to be monotonically non-decreasing. Additionally, this framework exhibits a distinctive capability to mitigate performance degradation caused by discrete quantization in practical systems, which is applicable to any algorithm intended for the continuous solution. Furthermore, as an integral component of the alternating iterations framework, we present a divide-and-sort (DaS) method to tackle the discrete inner product maximization problem. In the realm of $\ell _{\infty } $ -norm maximization, the DaS method ensures the identification of the global optimum with polynomial search complexity. We validate the proposed methods’ effectiveness and superiority through numerical and prototype experiments. Finally, we demonstrate that the proposed framework can be extended and applied to a wide range of other engineering problems.