Distributed Optimization over Lossy Networks via Relaxed Peaceman-Rachford Splitting: a Robust ADMM Approach

Abstract

In this work we address the problem of distributed optimization of the sum of convex cost functions in the context of multi-agent systems over lossy communication networks. Building upon operator theory, first, we derive an ADMMlike algorithm, referred to as relaxed ADMM (R-ADMM) via a generalized Peaceman-Rachford Splitting operator on the Lagrange dual formulation of the original optimization problem. This algorithm depends on two parameters, namely the averaging coefficient α and the augmented Lagrangian coefficient p and we show that by setting α = 1/2 we recover the standard ADMM algorithm as a special case. Moreover, first, we reformulate our R-ADMM algorithm into an implementation that presents reduced complexity in terms of memory, communication and computational requirements. Second, we propose a further reformulation which let us provide the first ADMM-like algorithm with guaranteed convergence properties even in the presence of lossy communication. Finally, this work is complemented with a set of compelling numerical simulations of the proposed algorithms over random geometric graphs subject to i.i.d. random packet losses.