Augmented Lagrangian methods as dynamical systems for constrained optimization

Dynamical systems approaches to constrained optimization often rely on a penalization term to reach feasible points, at the cost of slower convergence. However, we show one can construct a discrete-time system that maintains the same convergence guarantees without requiring such penalization term. We demonstrate that the sequential homotopy method, namely taking projected backward Euler steps on a projected gradient/anti-gradient augmented Lagrangian flow, matches with the classical augmented Lagrangian method without the multiplier estimate update. Then, we introduce a time-scaled flow and provide an interpretation of augmented Lagrangian methods as discrete-time dynamical systems. This approach inspires a simple yet effective method for nonlinear programming. We report on numerical results for equality-constrained problems.