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

Abstract

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 timevarying 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 PredictionCorrection 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.

A Coordinate-Descent Algorithm for Tracking Solutions in Time-Varying Optimal Power Flows

“…The network is based on a single-phase variant of the IEEE 37-node test case. It replaces constant load of 18 secondary transformers (at nodes 4,7,10,13,17,20,22,23,26,28,29,30,31,32,33,34,35, and 36, as highlighted in Figure 1) with real load data from Anatolia, California, sampled with 1 Hz frequency in August 2012 [2]. Further, the generation at PV plants is simulated based on real solar irradiance data in [2], with rating of these inverters at 300 kVA at node 3; 350 kVA at nodes 15, 16, and 200 kVA for all other inverters.…”

“…As the volatility of parameters of optimal power flows increases, there is a considerable interest in the pursuit of solutions to optimal power flows (OPF) in the on-line setting. In convex optimization and signal processing, related approaches are known as warm-starting [7,13,33], time-varying convex optimization [30], and dynamic convex optimization [29]. Much of the general-purpose work has, however, focussed on the use of interior-point methods [7,13,28], where a small number of computationally-demanding iterations suffice [14] to reach machine precision.…”

Optimal pricing strategy for resource allocation in 5G heterogeneous cellular networks

“…Hence, the KKT conditions are employed here to derive the optimal solutions. The Lagrangian associated with can be expressed as where λ c is the Lagrange multiplier.…”

“…Hence, the KKT conditions 29 are employed here to derive the optimal solutions. The Lagrangian associated with (15) can be expressed as…”

Dynamic Average Diffusion With Randomized Coordinate Updates

“…The references mentioned so far focus mainly on deriving solutions in the primal domain and use first-order distributed algorithms for tracking. There are of course other families of distributed methods for tracking dynamic signals, such as those based on the ADMM procedure [46,47], and the distributed Kalman filter [48][49][50]. These are powerful methods with good convergence rates.…”