Distributive Network Utility Maximization (NUM) over Time-Varying Fading Channels

TL;DR

This paper analyzes the convergence and tracking errors of a distributed primal-dual gradient algorithm for network utility maximization over time-varying fading channels, proposing an adaptive solution with improved performance.

Contribution

It investigates the behavior of the primal-dual scaled gradient algorithm under dynamic channels and proposes a low complexity adaptive scaling matrix method.

Findings

Converges to a limit region under Markov fading channels.

Tracking errors grow proportionally to T/N, with T as update interval and N as sojourn time.

Proposed adaptive algorithm outperforms baseline schemes.

Abstract

Distributed network utility maximization (NUM) has received an increasing intensity of interest over the past few years. Distributed solutions (e.g., the primal-dual gradient method) have been intensively investigated under fading channels. As such distributed solutions involve iterative updating and explicit message passing, it is unrealistic to assume that the wireless channel remains unchanged during the iterations. Unfortunately, the behavior of those distributed solutions under time-varying channels is in general unknown. In this paper, we shall investigate the convergence behavior and tracking errors of the iterative primal-dual scaled gradient algorithm (PDSGA) with dynamic scaling matrices (DSC) for solving distributive NUM problems under time-varying fading channels. We shall also study a specific application example, namely the multi-commodity flow control and multi-carrier power allocation problem in multi-hop ad hoc networks. Our analysis shows that the PDSGA converges to a limit region rather than a single point under the finite state Markov chain (FSMC) fading channels. We also show that the order of growth of the tracking errors is given by O(T/N), where T and N are the update interval and the average sojourn time of the FSMC, respectively. Based on this analysis, we derive a low complexity distributive adaptation algorithm for determining the adaptive scaling matrices, which can be implemented distributively at each transmitter. The numerical results show the superior performance of the proposed dynamic scaling matrix algorithm over several baseline schemes, such as the regular primal-dual gradient algorithm.