Passivity-based interpretation of the Tracking-ADMM algorithm for distributed constraint-coupled optimization

We propose a system-theoretic perspective on Tracking-ADMM, a recently introduced distributed optimization algorithm for constraint-coupled optimization problems that combines ADMM and tracking over networks. Through a judicious coordinate transformation, we formulate the algorithm as a dynamical linear component in closed-loop with a static nonlinearity resulting from an optimization step. We show, by direct calculation, that the linear component is discrete-positive real, while the static nonlinearity is monotone. As both components are passive systems, their feedback interconnection is passive as well, thus ensuring the stability of Tracking-ADMM.