Optimal Discrete Beamforming of RIS-Aided Wireless Communications: An Inner Product Maximization Approach

This paper addresses the combinatorial optimization challenges arising in practical communication services employing reconfigurable intelligent surfaces (RISs) with discrete phase configurations. Our primary focus is on achieving optimal beamforming for power enhancement in RIS-aided wireless communications, a versatile task applicable to various channel assumptions. Traditional approaches often require resource-intensive exponential search techniques and fall into the universal (NP-hard) category. We formally define this task as a discrete inner product maximization problem. By leveraging the inherent structure of this problem, we introduce an equivalent yet separable formulation, which consists of computation-friendly subproblems. A key benefit of this reformulation is that it significantly reduces the exponential search space, resulting in polynomial search complexity. We propose a highly efficient divide-and-sort (DaS) search framework designed to ensure the achievement of global optima for the original discrete inner product maximization problem. Notably, the final algorithm exhibits linear search complexity in terms of the number of discrete variables, making it particularly effective for large-scale problems. Our comprehensive numerical studies and real-world prototype experiments demonstrate the efficiency and effectiveness of the proposed DaS search algorithm. Additionally, we establish that, for moderate-resolution quantization (4-bit and above), there is no noticeable difference in power gains between continuous and discrete phase configurations.