  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. 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. 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.
