  For a class of nonsmooth composite optimization problems with linear equality
constraints, we utilize a Lyapunov-based approach to establish the global
exponential stability of the primal-dual gradient flow dynamics based on the
proximal augmented Lagrangian. The result holds when the differentiable part of
the objective function is strongly convex with a Lipschitz continuous gradient;
the non-differentiable part is proper, lower semi-continuous, and convex; and
the matrix in the linear constraint is full row rank. Our quadratic Lyapunov
function generalizes recent result from strongly convex problems with either
affine equality or inequality constraints to a broader class of composite
optimization problems with nonsmooth regularizers and it provides a worst-case
lower bound of the exponential decay rate. Finally, we use computational
experiments to demonstrate that our convergence rate estimate is less
conservative than the existing alternatives.
