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 have some troubles.The problem with regard to $\mathbf u =[u_1,\cdots,u_n]$ variables is formulated as follow my Lagrangian as the form$$\mathcal L(\mathbf u,\lambda):= \mathcal W(\mathbf u) + \rho \lambda^T \,c(\mathbf u) + \frac{\rho}{2} c^2(\mathbf u) $$So generally The goal at each Newton iteration is to solve a problem of the form$$A \Delta u = b$$With (we drop hessian of the constraint)$$A(\mathbf u,\rho): =  \nabla_{\mathbf u}^2 \mathcal W(\mathbf u) + \rho C^T(\mathbf u)C(\mathbf u) $$ and$$b(\mathbf u,\rho) :=- \big(\nabla_{\mathbf u}\mathcal W(\mathbf u) +(\rho +\lambda^Tc(\mathbf u)) \nabla_{\mathbf u}(\mathbf u)\big)$$and the capital $C$ is meant for $C(\mathbf u) := \nabla_{\mathbf u} c(\mathbf u)$.Thank you.
Efficient preconditioner for Augmented Lagrangian
linear solver;preconditioning;constrained optimization