Semi-Global Exponential Stability of Augmented Primal-Dual Gradient Dynamics for Constrained Convex Optimization
For researchers in optimization and control, this work provides a theoretical guarantee for a widely used method, though it is an incremental extension of existing results.
This paper extends augmented primal-dual gradient dynamics to handle general convex and nonlinear inequality constraints, proving semi-global exponential stability for strongly convex objectives. It also provides a counterexample showing that global exponential stability is not always achieved.
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.