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In this paper, we propose (i) a novel distributed algorithm for consensusoptimization over networks and (ii) a robust extension tailored to deal withasynchronous agents and packet losses. The key idea is to achieve dynamicconsensus on (i) the agents' average and (ii) the global descent direction byiteratively solving an online auxiliary optimization problem through adistributed implementation of the Alternating Direction Method of Multipliers(ADMM). Such a mechanism is suitably interlaced with a local proportionalaction steering each agent estimate to the solution of the original consensusoptimization problem. First, in the case of ideal networks, by using tools fromsystem theory, we prove the linear convergence of the scheme with stronglyconvex costs. Then, by exploiting the averaging theory, we extend such a firstresult to prove that the robust extension of our method preserves linearconvergence in the case of asynchronous agents and packet losses. Further, byusing the notion of Input-to-State Stability, we also guarantee the robustnessof the schemes with respect to additional, generic errors affecting the agents'updates. Finally, some numerical simulations confirm our theoretical findingsand show that the proposed methods outperform the existing state-of-the-artdistributed methods for consensus optimization.

ADMM-Tracking Gradient for Distributed Optimization over Asynchronous and Unreliable Networks