  We develop algorithms that find and track the optimal solution trajectory of
time-varying convex optimization problems which consist of local and
network-related objectives. The algorithms are derived from the
prediction-correction methodology, which corresponds to a strategy where the
time-varying problem is sampled at discrete time instances and then a sequence
is generated via alternatively executing predictions on how the optimizers at
the next time sample are changing and corrections on how they actually have
changed. Prediction is based on how the optimality conditions evolve in time,
while correction is based on a gradient or Newton method, leading to
Decentralized Prediction-Correction Gradient (DPC-G) and Decentralized
Prediction-Correction Newton (DPC-N). We extend these methods to cases where
the knowledge on how the optimization programs are changing in time is only
approximate and propose Decentralized Approximate Prediction-Correction
Gradient (DAPC-G) and Decentralized Approximate Prediction-Correction Newton
(DAPC-N). Convergence properties of all the proposed methods are studied and
empirical performance is shown on an application of a resource allocation
problem in a wireless network. We observe that the proposed methods outperform
existing running algorithms by orders of magnitude. The numerical results
showcase a trade-off between convergence accuracy, sampling period, and network
communications.
