  The Primal-Dual (PD) algorithm is widely used in convex optimization to
determine saddle points. While the stability of the PD algorithm can be easily
guaranteed, strict contraction is nontrivial to establish in most cases. This
work focuses on continuous, possibly non-autonomous PD dynamics arising in a
network context, in distributed optimization, or in systems with multiple
time-scales. We show that the PD algorithm is indeed strictly contracting in
specific metrics and analyze its robustness establishing stability and
performance guarantees for different approximate PD systems. We derive
estimates for the performance of multiple time-scale multi-layer optimization
systems, and illustrate our results on a primal-dual representation of the
Automatic Generation Control of power systems.
