Optimal Local Precoding for Cell-Free Massive MIMO Under the UatF Bound

This paper investigates the optimal local precoding design for cell-free massive MIMO systems, subject to per-user quality-of-service (QoS) constraints and per-access-point (AP) power limits. In this distributed setting, each AP determines its precoding strategy using only locally acquired instantaneous channel state information (CSI) alongside globally available statistical CSI. Since the precoding design based on the use-and-then-forget (UatF) bound involves expectation terms, its optimization problem should be formulated as a functional optimization problem. Unlike function optimization problems whose solutions are numerical precoding values, the solution to a functional optimization problem is a precoding function. Consequently, each AP computes its precoding vectors in real-time by applying this derived optimal function to its locally acquired instantaneous CSI. By applying functional optimization theory and the Karush-Kuhn-Tucker (KKT) conditions, we rigorously prove that the optimal local precoder is structurally equivalent to a Local Minimum Mean Square Error (L-MMSE) precoder, with undetermined Lagrange multiplier parameters associated. We then derive an efficient iterative algorithm to compute these multipliers and establish its convergence to a unique fixed point that yields the optimal precoding solution, thereby providing a complete framework for optimal local downlink precoding. Simulation results demonstrate that the proposed method achieves substantial gains over state-of-the-art approaches, validating its effectiveness and robustness across various system configurations.