A note on the convergence of an augmented Lagrangian algorithm to second-order stationary points

Abstract

Many algorithms that ensure second-order necessary optimality conditions were developed in the literature. To the best of our knownledge, none of them guarantee Strong Second-Order Necessary Condition (SSONC). Gould and Toint [5] showed that we do not expect SSONC in the barrier method. In this paper, we argue by an example that the same is true for the second-order augmented Lagrangian method introduced in [1]. This reinforces the Weak Second-Order Necessary Condition as the appropriate condition for the convergence analysis of second-order optimization algorithms.