Enhancing the Downlink Rate Fairness of Low-Resolution Active RIS-Aided Signaling by Closed-Form Expression-Based Iterative Optimization

This paper proposes a joint design strategy for enhancing individual user rates in a multi-user system by optimizing both the programmable reflecting elements (PREs) of an active reconfigurable intelligent surface (aRIS) and the transmit beamforming at a base station. Given that the aRIS's PREs are bound by discrete constraints due to low-resolution quantization, this design approach relies on large-scale mixed discrete-continuous problems, which are addressed through a new universal penalised optimization reformulations. Initially, we develop iterations based on convex quadratic solvers (CQ) to tackle the problem of maximizing the users' minimum rate (MR). Given that the computational complexity of these CQs is cubic, leading to high costs in large-scale computations, we introduce a pair of surrogate objectives. These objectives are designed in a way that their constrained optimization can be efficiently managed through iterations of closed-form expressions with scalable complexity, rendering them practical for large-scale computations. This pair of surrogate objectives comprises the maximization of the geometric mean of users' rates (GM-rate maximization) and the soft-maximization of users' MR (soft max-min rate optimization). Remarkably, they not only enhance MR but also contribute to the improvement of the sum-rate (SR). Building upon the GM-rate optimization, we further propose addressing the energy efficiency problem, which achieves a high ratio of SR to power consumption and MR to power dissipation through closed-form expressions. Comprehensive simulations are conducted to validate the efficacy of the proposed solutions.