Tight Differentiable Relaxation of Sum Ergodic Capacity Maximization for Clustered Cell-Free Networking

Clustered cell-free networking emerges as a promising paradigm for future communication systems. To optimize clustered cell-free networking, we aim to maximize the sum ergodic capacity, which stands as a key performance metric of wireless communications systems. However, the networking aspect leads to a difficult combinatorial problem. Moreover, the implicitness of ergodic capacity results in obstacles to evaluation and optimization. Existing works often rely on inaccurate approximations of ergodic capacity and/or high-cost integer programming (IP) algorithms, rendering them impractical for real-world deployment. In this paper, we formulate the problem as an IP problem and relax it into a continuous one with the ergodic capacity approximated by its deterministic equivalents. Importantly, we establish the tightness of the relaxation without sacrificing the optimality of the solution. We then employ Breg-man proximal gradient (BPG) nested with Dykstra's algorithm to solve the relaxed problem and show the convergence for both BPG and its subproblems. Simulation results verify the effectiveness and efficiency of our approach.