Decentralized Prediction-Correction Methods for Networked Time-Varying Convex Optimization

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 methods we propose 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 keeping track of the residual dynamics of the optimality conditions, 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 also 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). These methods use a first-order backward approximation to estimate the time variation of the functions. We also study the convergence properties of the proposed methods. We next show an application of a resource allocation problem in a wireless network, and observe that the proposed methods outperform existing running algorithms by orders of magnitude. Moreover, numerical results showcase a trade-off between convergence accuracy, sampling period, and network communications.