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In this work we address the problem of distributed optimization of the sum ofconvex cost functions in the context of multi-agent systems over lossycommunication networks. Building upon operator theory, first, we derive anADMM-like algorithm that we refer to as relaxed ADMM (R-ADMM) via a generalizedPeaceman-Rachford Splitting operator on the Lagrange dual formulation of theoriginal optimization problem. This specific algorithm depends on twoparameters, namely the averaging coefficient $\alpha$ and the augmentedLagrangian coefficient $\rho$. We show that by setting $\alpha=1/2$ we recoverthe standard ADMM algorithm as a special case of our algorithm. Moreover, byproperly manipulating the proposed R-ADMM, we are able to provide twoalternative ADMM-like algorithms that present easier implementation and reducedcomplexity in terms of memory, communication and computational requirements.Most importantly the latter of these two algorithms provides the firstADMM-like algorithm which has guaranteed convergence even in the presence oflossy communication under the same assumption of standard ADMM with losslesscommunication. Finally, this work is complemented with a set of compellingnumerical simulations of the proposed algorithms over cycle graphs and randomgeometric graphs subject to i.i.d. random packet losses.

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