On the Exponential Stability of Primal-Dual Gradient Dynamics

Continuous time primal-dual gradient dynamics (PDGD) that find a saddle point of a Lagrangian of an optimization problem have been widely used in systems and control. While the global asymptotic stability of such dynamics has been well-studied, it is less studied whether they are globally exponentially stable. In this letter, we study the PDGD for convex optimization with strongly convex and smooth objectives and affine equality or inequality constraints, and prove global exponential stability for such dynamics. Bounds on decaying rates are provided.