Global exponential stability of primal dual gradient flow dynamics based on the proximal augmented Lagrangian: A Lyapunov based approach
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.
2020 59th IEEE Conference on Decision and Control (CDC)