On Saddle Points of Augmented Lagrangians for Constrained Nonconvex Optimization

We present in this paper new results on the existence of saddle points of augmented Lagrangian functions for constrained nonconvex optimization. Four classes of augmented Lagrangian functions are considered: the essentially quadratic augmented Lagrangian, the exponential-type augmented Lagrangian, the modified barrier augmented Lagrangian, and the penalized exponential-type augmented Lagrangian. We first show that under second-order sufficiency conditions, all these augmented Lagrangian functions possess local saddle points. We then prove that global saddle points of these augmented Lagrangian functions exist under certain mild additional conditions. The results obtained in this paper provide a theoretical foundation for the use of augmented Lagrangians in constrained global optimization. Our findings also give new insights to the role played by augmented Lagrangians in local duality theory of constrained nonconvex optimization.