Contractivity of Distributed Optimization and Nash Seeking Dynamics

TL;DR

This paper analyzes the contractivity properties of distributed optimization and Nash equilibrium dynamics using contraction theory, providing bounds, conditions for strong contractivity, and insights into game-theoretic scenarios.

Contribution

It introduces a novel bound on the logarithmic norm of saddle matrices and establishes strong contractivity conditions for various distributed and game-theoretic dynamics.

Findings

Bound on the logarithmic norm of saddle matrices.

Strong contractivity of distributed gradient flows over arbitrary topologies.

Equivalent conditions for contractivity in pseudogradient and best response games.

Abstract

In this letter, we study distributed optimization and Nash equilibrium-seeking dynamics from a contraction theoretic perspective. Our first result is a novel bound on the logarithmic norm of saddle matrices. Second, for distributed gradient flows based upon incidence and Laplacian constraints over arbitrary topologies, we establish strong contractivity over an appropriate invariant vector subspace. Third, we give sufficient conditions for strong contractivity in pseudogradient and best response games with complete information, show the equivalence of these conditions, and consider the special case of aggregative games.