  We revisit an algorithm for distributed consensus optimization proposed in
2010 by J. Wang and N. Elia. By means of a Lyapunov-based analysis, we prove
input-to-state stability of the algorithm relative to a closed invariant set
composed of optimal equilibria and with respect to perturbations affecting the
algorithm's dynamics. In the absence of perturbations, this result implies
linear convergence of the local estimates and Lyapunov stability of the optimal
steady state. Moreover, we unveil fundamental connections with the well-known
Gradient Tracking and with distributed integral control. Overall, our results
suggest that a control theoretic approach can have a considerable impact on
(distributed) optimization, especially when robustness is considered.
