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

TL;DR

This paper introduces decentralized prediction-correction algorithms for tracking solutions of time-varying convex optimization problems in networks, demonstrating superior empirical performance over existing methods.

Contribution

The paper proposes novel decentralized prediction-correction algorithms, including approximate variants, with proven convergence properties for time-varying convex optimization.

Findings

Algorithms outperform existing methods by orders of magnitude.

Trade-offs exist between convergence accuracy, sampling period, and network communication.

Empirical results validate the effectiveness of the proposed methods in wireless network resource allocation.

Abstract

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.