On the Q-linear convergence of Distributed Generalized ADMM under non-strongly convex function components

TL;DR

This paper proves that Generalized Distributed ADMM converges Q-linearly for convex functions in multi-agent optimization when the overall objective is strongly convex, even if individual functions are only convex, and explores its relation to P-EXTRA.

Contribution

It establishes Q-linear convergence of Distributed Generalized ADMM under weak conditions and shows its equivalence to P-EXTRA for certain matrices, providing new insights.

Findings

Q-linear convergence under weak convexity assumptions

Equivalence between Generalized Distributed ADMM and P-EXTRA

Insights into accelerated convergence with overshooting

Abstract

Solving optimization problems in multi-agent networks where each agent only has partial knowledge of the problem has become an increasingly important problem. In this paper we consider the problem of minimizing the sum of $n$ convex functions. We assume that each function is only known by one agent. We show that Generalized Distributed ADMM converges Q-linearly to the solution of the mentioned optimization problem if the over all objective function is strongly convex but the functions known by each agent are allowed to be only convex. Establishing Q-linear convergence allows for tracking statements that can not be made if only R-linear convergence is guaranteed. Further, we establish the equivalence between Generalized Distributed ADMM and P-EXTRA for a sub-set of mixing matrices. This equivalence yields insights in the convergence of P-EXTRA when overshooting to accelerate convergence.