  This paper analyzes the contraction of the primal-dual gradient optimization
via contraction theory in the context of discrete-time updating dynamics. The
contraction theory based on Riemannian manifolds is first established for
convergence analysis of a convex optimization algorithm. The equality and
inequality constrained optimization cases are studied, respectively. Under some
reasonable assumptions, we construct the Riemannian metric to characterize a
contraction region. It is shown that if the step-sizes of the updating dynamics
are properly designed, the convergence rates for both cases can be obtained
according to the contraction region in which the convergence can be guaranteed.
Moreover, the augmented Lagrangian function which is projection free is adopted
to tackle the inequality constraints. Some numerical experiments are simulated
to demonstrate the effectiveness of the presented contraction analysis results
on primal-dual gradient optimization algorithm.
