A Probabilistic Reformulation Technique for Discrete RIS Optimization in Wireless Systems

The use of reconfigurable intelligent surfaces (RIS) can improve wireless communication by modifying the wireless link to create virtual line-of-sight links, bypass blockages, suppress interference, and enhance localization. However, enabling the RIS to modify the wireless channel requires careful optimization of the RIS phase-shifts. Although discrete RIS is more practical given hardware limitations, continuous RIS phase-shift optimization has attracted significantly more attention than discrete RIS optimization, which suffers from issues like quantization error and scalability. To overcome these issues, we develop a comprehensive probabilistic technique to transform discrete optimization problems into optimization problems of continuous domain probability parameters by interpreting the discrete optimization variable as a categorical random vector and computing expectations with respect to those parameters. We rigorously establish that for the unconstrained case, the optimal points of the reformulation and the original problem coincide. For the constrained case, we prove that the transformed problem is a relaxation of the original problem. We apply the proposed technique to two canonical discrete RIS applications: SINR maximization and overhead-aware rate and energy efficiency (EE) maximization. The reformulation enables both stochastic and analytical interpretations of the original problems, as we demonstrate in our RIS applications. The former interpretation yields a stochastic sampling technique, whereas the latter yields an analytical gradient descent (GD) approach that employs closed-form approximations for the expectation. The numerical results demonstrate that the proposed technique is applicable to a variety of discrete RIS optimization problems and outperforms other general approaches, such as closest point projection (CPP) and semidefinite relaxation (SDR) methods.