Semi-global exponential stability of augmented 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 (Aug-PDGD) to incorporate general convex and nonlinear inequality constraints, and we establish its semi-global exponential stability when the objective function is strongly convex. We also provide an example of a strongly convex quadratic program of which the Aug-PDGD fails to achieve global exponential stability. Numerical simulation also suggests that the exponential convergence rate could depend on the initial distance to the KKT point.