  In this paper, we propose a novel distributed algorithm for consensus
optimization over networks and a robust extension tailored to deal with
asynchronous agents and packet losses. Indeed, to robustly achieve dynamic
consensus on the solution estimates and the global descent direction, we embed
in our algorithms a distributed implementation of the Alternating Direction
Method of Multipliers (ADMM). Such a mechanism is suitably interlaced with a
local proportional action steering each agent estimate to the solution of the
original consensus optimization problem. First, in the case of ideal networks,
by using tools from system theory, we prove the linear convergence of the
scheme with strongly convex costs. Then, by exploiting the averaging theory, we
extend such a first result to prove that the robust extension of our method
preserves linear convergence in the case of asynchronous agents and packet
losses. Further, by using the notion of Input-to-State Stability, we also
guarantee the robustness of the schemes with respect to additional, generic
errors affecting the agents' updates. Finally, some numerical simulations
confirm our theoretical findings and compare our algorithms with other
distributed schemes in terms of speed and robustness.
