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

Abstract

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.

References in corpus (4)

• On the Exponential Stability of Primal-Dual Gradient Dynamics
• The proximal augmented Lagrangian method for nonsmooth composite optimization
• On the Exponential Stability of Projected Primal-Dual Dynamics on a Riemannian Manifold
• Global exponential stability of primal-dual gradient flow dynamics based on the proximal augmented Lagrangian: A Lyapunov-based approach

Cited by in corpus (4)

• Aug-PDG: Linear Convergence of Convex Optimization with Inequality Constraints
• Global exponential stability of primal-dual gradient flow dynamics based on the proximal augmented Lagrangian: A Lyapunov-based approach
• On the Geometry and Linear Convergence of Primal-Dual Dynamics
• Exponential Stability of Partial Primal-Dual Gradient Dynamics with Nonsmooth Objective Functions