On the Dual-Decomposition-Based Sum Capacity Maximization for Vector Broadcast Channels

The dual-decomposition-based sum capacity maximization algorithm for the vector broadcast channels consists of the following two nested loops: The inner loop is an iterative water-filling-like algorithm that optimizes the covariance matrices for a fixed water level, and the outer loop finds the optimum water level by using a bisection search. Recently, an improved version has been proposed, where the inner loop is optimized only to a certain tolerance that still guarantees algorithm convergence. This requires tracking the instantaneous error of the inner loop, and this issue has not yet been addressed. In this paper, we provide a tight upper bound of the instantaneous error of the inner loop, which can be used to construct a nonheuristic stopping criterion. Specifically, we derive the Lagrange dual function of the inner optimization problem and provide a method to obtain a feasible dual variable from which a tight upper bound of the instantaneous error can easily be computed. We also show that the rate of convergence of the global algorithm can substantially be improved by introducing a covariance normalization step between two successive outer iterations.