Efficient preconditioner for Augmented Lagrangian

tlaverne's picture

Hi folks !

I have a question which is more mathematical than mechanical. However since it is to solve mechanical problem, one of you may have an answer !

I want to solve a non-linear problem with non-linear equality constrains and I'm using a augmented Lagrangian with a penalty regularization term that, as well known, spoils the condition number of my linearized systems (at each Newton iteration I mean). The bigger the penalty term, the worse the condition number is. Would someone know an efficient way to get rid of this bad conditioning in that specific case ?

To be more specific, I'm using the classical augmented lagrangian because I have lots of constraints which may generally be redundant. So blindly incorporating the constraints direclty into the primal variables is very convenient. I tried other more sophisticated approaches based on variable eliminations or efficient preconditioners directly on the KKT system but, because of constraints redundancy, I had some troubles.

mohamedlamine's picture

Re : Efficient Preconitionner

Hi tlaverne,

Newton's Methods may not converge in some cases : They depend on the choice of the starting point. I recommend in such cases to use Conjugate Gradient Method or Efficiently the Preconitionned Conjugate Gradient Method with C=L*Lt as a preconitionner. If your system of equations is Linear L can be obtained from L*U factorization of the main matrix.



Mohammed Lamine

re: Efficient Preconditioner

Perhaps you might want to ask on the petsc mailing list. 


The LNKS papers by George Biros, Omar Ghattas et al also might be useful.  



tlaverne's picture

Mohammed Lamine thank you

Mohammed Lamine
thank you for your answer but I'm not really looking for a preconditioner by itself, rather a method to reduce the ill-conditioning of my system. Sorry if my question was confusing. Since what I'm doing is pretty standard I though there might be some standard way to avoid ill-conditoning.


Nachiket, at the best of my knowledge LNKS method use some decomposition of the variables into two sets: on set of master variables and one set of slave variables (or state and decision variables in their paper). I have absolutely no idea to do that on my problem easily except doing a sparse QR on the gradient of my constraint. Might be the way to go, but I don't have much working knowledge about sparse QR factorization efficiency.

My blog on research on Hybrid Solvers: http://mechenjoy.blogspot.com/