A Contraction Analysis of Primal-Dual Dynamics in Distributed and Time-Varying Implementations

Abstract

In this article, we provide an overarching analysis of primal-dual dynamics associated with linear equality-constrained optimization problems using contraction analysis. For the well-known standard version of the problem, we establish convergence under convexity and the contracting rate under strong convexity. Then, for a canonical distributed optimization problem, we use partial contractivity to establish global exponential convergence of its primal-dual dynamics. As an application, we propose a new distributed solver for the least-squares problem with the same convergence guarantees. Finally, for time-varying versions of both centralized and distributed primal-dual dynamics, we exploit their contractive nature to establish bounds on their tracking error. To support our analyses, we introduce novel results on contraction theory.