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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. 

We consider a class of nonsmooth convex composite optimization problems, where the objective function is given by the sum of a continuously differentiable convex term and a potentially non-differentiable convex regularizer. In [1], the authors introduced the proximal augmented Lagrangian method and derived the resulting continuous-time primal-dual dynamics that converge to the optimal solution. In this paper, we extend these dynamics from continuous to discrete time via the forward Euler discretization. We prove explicit bounds on the exponential convergence rates of our proposed algorithm with a sufficiently small step size. Since a larger step size can improve the convergence speed, we further develop a linear matrix inequality (LMI) condition which can be numerically solved to provide rate certificates with general step size choices. In addition, we prove that a large range of step size values can guarantee exponential convergence. We close the paper by demonstrating the performance of the proposed algorithm via computational experiments.