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

• Published in: arXiv.org
• Main Authors: Bastianello, Nicola, Todescato, Marco, Carli, Ruggero, Schenato, Luca
• Format: Paper Journal Article
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 26.09.2018
• Subjects: Algorithms; Communication; Communication networks; Computer memory; Computer simulation; Graphs; Mathematics - Optimization and Control; Multiagent systems; Operators (mathematics); Optimization; Robustness (mathematics); Splitting
• ISSN: 2331-8422
• DOI: 10.48550/arxiv.1809.09887

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 ADMM-like algorithm that we refer to as relaxed ADMM (R-ADMM) via a generalized Peaceman-Rachford Splitting operator on the Lagrange dual formulation of the original optimization problem. This specific algorithm depends on two parameters, namely the averaging coefficient \(\alpha\) and the augmented Lagrangian coefficient \(\rho\). We show that by setting \(\alpha=1/2\) we recover the standard ADMM algorithm as a special case of our algorithm. Moreover, by properly manipulating the proposed R-ADMM, we are able to provide two alternative ADMM-like algorithms that present easier implementation and reduced complexity in terms of memory, communication and computational requirements. Most importantly the latter of these two algorithms provides the first ADMM-like algorithm which has guaranteed convergence even in the presence of lossy communication under the same assumption of standard ADMM with lossless communication. Finally, this work is complemented with a set of compelling numerical simulations of the proposed algorithms over cycle graphs and random geometric graphs subject to i.i.d. random packet losses.