Semi-Global Exponential Stability of Primal-Dual Gradient Dynamics for Constrained Convex Optimization.

Primal-dual gradient dynamics that find saddle points of a Lagrangian have been widely employed for handling constrained optimization problems. Building on existing methods, we extend the augmented primal-dual gradient dynamics to incorporate general convex and nonlinear inequality constraints, and we establish its semi-global exponential stability when the objective function has a quadratic gradient growth. Numerical simulation also suggests that the exponential convergence rate could depend on the initial distance to the KKT point.