Global exponential stability of primal-dual gradient flow dynamics based on the proximal augmented Lagrangian

The primal-dual gradient flow dynamics based on the proximal augmented Lagrangian were introduced in [1] to solve nonsmooth composite optimization problems with a linear equality constraint. We use a Lyapunov-based approach to demonstrate global exponential stability of the underlying dynamics when the differentiable part of the objective function is strongly convex and its gradient is Lipschitz continuous. This also allows us to determine a bound on the stepsize that guarantees linear convergence of the discretized algorithm.