In this letter, the achievable rate maximization problem is considered for intelligent reflecting surface (IRS) assisted multiple-input multiple-output (MIMO) systems in an underlay spectrum sharing scenario, subject to interference power constraints at the 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 grows linearly with respect to the number of reflecting elements at the IRS, while that of the two benchmark methods grows with the third power of the number of IRS elements. Moreover, numerical results show that the proposed PDDGP algorithm yields considerably higher achievable rate than the benchmark solutions.
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