Exponential Convergence of Primal-Dual Dynamics Under General Conditions and Its Application to Distributed Optimization.

Abstract

In this article, we establish the local and global exponential convergence of a primal-dual dynamics (PDD) for solving equality-constrained optimization problems without strong convexity and full row rank assumption on the equality constraint matrix. Under the metric subregularity of Karush-Kuhn-Tucker (KKT) mapping, we prove the local exponential convergence of the dynamics. Moreover, we establish the global exponential convergence of the dynamics in an invariant subspace under a technically designed condition which is weaker than strong convexity. As an application, the obtained theoretical results are used to show the exponential convergence of several existing state-of-the-art primal-dual algorithms for solving distributed optimization without strong convexity. Finally, we provide some experiments to demonstrate the effectiveness of our results.