Differential equation method based on approximate augmented Lagrangian for nonlinear programming

This paper analyzes the approximate augmented Lagrangian dynamical systems for constrained optimization. We formulate the differential systems based on first derivatives and second derivatives of the approximate augmented Lagrangian. The solution of the original optimization problems can be obtained at the equilibrium point of the differential equation systems, which lead the dynamic trajectory into the feasible region. Under suitable conditions, the asymptotic stability of the differential systems and local convergence properties of their Euler discrete schemes are analyzed, including the locally quadratic convergence rate of the discrete sequence for the second derivatives based differential system. The transient behavior of the differential equation systems is simulated and the validity of the approach is verified with numerical experiments.