A System Theoretical Perspective to Gradient-Tracking Algorithms for Distributed Quadratic Optimization

TL;DR

This paper applies system theory to analyze a gradient-tracking distributed optimization algorithm for quadratic problems, revealing structural properties and stability conditions within a linear system framework.

Contribution

It introduces a novel system-theoretic framework to analyze the structural and stability properties of gradient-tracking algorithms in distributed quadratic optimization.

Findings

The algorithm can be modeled as a sparse closed-loop linear system.

Asymptotic stability is limited to a proper invariant set.

Global convergence requires proper initialization.

Abstract

In this paper we consider a recently developed distributed optimization algorithm based on gradient tracking. We propose a system theory framework to analyze its structural properties on a preliminary, quadratic optimization set-up. Specifically, we focus on a scenario in which agents in a static network want to cooperatively minimize the sum of quadratic cost functions. We show that the gradient tracking distributed algorithm for the investigated program can be viewed as a sparse closed-loop linear system in which the dynamic state-feedback controller includes consensus matrices and optimization (stepsize) parameters. The closed-loop system turns out to be not completely reachable and asymptotic stability can be shown restricted to a proper invariant set. Convergence to the global minimum, in turn, can be obtained only by means of a proper initialization. The proposed system interpretation of the distributed algorithm provides also additional insights on other structural properties and possible design choices that are discussed in the last part of the paper as a starting point for future developments.