On the Augmented Lagrangian

Abstract
It is well known that, for a nonlinear program, any Karush–Kuhn–Tucker point that satisfies the second order sufficient condition is also a strict local minimum of the augmented Lagrangian with an underlying penalty parameter sufficiently large. A main theme of the paper is to strengthen the theorem and bring to light an interesting connection among general critical points of a nonlinear program and its augmented Lagrangian. When a penalty parameter is sufficiently large, it will be shown that a Karush–Kuhn–Tucker point of a nonlinear program is also a critical point of its augmented Lagrangian with the same index and the same nullity. The aforementioned theorem is just the special case when the index and the nullity are zeros. This result is established through an extension of a lemma due to Finsler and Debreu, which is interesting in its own right. Consequently, under a constraint qualification, any Karush–Kuhn–Tucker point can also be shown to satisfy a second order necessary condition for a local maximum of the generalized Wolfe dual induced from the augmented Lagrangian. Interestingly, this relation holds without requiring the Karush–Kuhn–Tucker point itself to satisfy any second order optimality condition.

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