I understand that augmented Lagrangian methods, add penalty terms to standard Lagrangian method.
The question is what is wrong with original standard Lagrangian method, that made people add a quadratic penalty term.
This is not a deep question, I am asking a very basic question. Do not see the motivation yet.
The motivation is that numerical optimization methods usually find an extremum of a function. But for constrained extrema which one wants to find with the Lagrangian method one has to find the stationary points of Lagrangian. The augmented Lagrangian method transforms stationary points of the Lagrangian into extrema of the augmented Lagrangian (if the parameters are chosen large enough). So if you can solve the problem analytically, no need for augmented L., otherwise in numerical solutions you might use them.