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. 

http://www.mcs.anl.gov/petsc/miscellaneous/mailing-lists.html

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

-Nachiket 

 


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/