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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.

user25004
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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.

Karl
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  • If we can NOT find the stationary points of Lagrangian ANALYTICALLY, can't we numerically and iteratively solve that? Why augment? – user25004 Feb 23 '15 at 19:52
  • Fine, if you can do this without using augmented Lagrangian methods or other helps , you will become famous. – Karl Feb 23 '15 at 19:59
  • :D What I hear from you is that, if we do not add the quadratic terms, but try to repeat the same procedure, (sth like optimize for primal, fix value of the primal, optimize for dual, alternate till convergence) something will go wrong. Right? – user25004 Feb 23 '15 at 20:13
  • Maybe you look at Nocedal's book and think yourself a little bit about the problem. I tried to gave you an answer with which you are not content as I see. – Karl Feb 23 '15 at 20:21