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 two nested loops: the inner loop is an iterative water-filling like algorithm which optimizes the covariance matrices for a fixed water level, and the outer loop is a bisection method which searches for the optimum water level. Recently, an improved version has been proposed, where the inner loop is optimized only to a certain tolerance that still guarantees the 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 non- heuristic 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 be easily computed.