Achievable Rate Maximization for Underlay Spectrum Sharing MIMO System with Intelligent Reflecting Surface

We consider the achievable rate maximization problem for intelligent reflecting surface (IRS) assisted multiple-input multiple-output systems in an underlay spectrum sharing scenario, subject to interference power constraints at primary users. The formulated non-convex optimization problem is challenging to solve due to its non-convexity as well as coupling design variables in the constraints. Different from existing works that are mostly based on alternating optimization (AO), we propose a penalty dual decomposition based gradient projection (PDDGP) algorithm to solve this problem. We also provide a convergence proof and a complexity analysis for the proposed algorithm. We benchmark the proposed algorithm against two known solutions, namely a minimum mean-square error based AO algorithm and an inner approximation method with block coordinate descent. Specifically, the complexity of the proposed algorithm is $O(N_I^2)$ while that of the two benchmark methods is $O(N_I^3)$, where $N_I$ is the number of IRS elements. Moreover, numerical results show that the proposed PDDGP algorithm yields considerably higher achievable rate than the benchmark solutions.