  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.
