Convergence Analysis of Saddle Point Problems in Time Varying Wireless Systems— Control Theoretical Approach

Saddle point problems arise from many wireless applications, and primal-dual iterative algorithms are widely applied to find the saddle points. In the existing literature, the convergence results of such algorithms are established assuming the problem specific parameters remain unchanged during the iterations. However, this assumption is unrealistic in time varying wireless systems, as explicit message passing is usually involved in the iterations and the channel state information (CSI) may change in a time scale comparable to the algorithm update period. This paper investigates the convergence behavior and the tracking error of primal-dual iterative algorithms under time varying CSI. The convergence results are established by studying the stability of an equivalent virtual dynamic system derived in the paper, and the Lyapunov theory is applied for the stability analysis. We show that the average tracking error is proportional to the time variation rate of the CSI. Based on these analyses, we also derive an adaptive primal-dual algorithm by introducing a compensation term to reduce the tracking error under the time varying CSI.