An example comparing the standard and safeguarded augmented Lagrangian methods

We consider the well-known augmented Lagrangian method for constrained optimization and compare its classical variant to a modified counterpart which uses safeguarded multiplier estimates. In particular, we give a brief overview of the theoretical properties of both methods, focusing on both feasibility and optimality of limit points. Finally, we give an example which illustrates the advantage of the modified method and incidentally shows that some of the assumptions used for convergence of the classical method cannot be relaxed.