  A preconditioning strategy for the Powell-Hestenes-Rockafellar Augmented
Lagrangian method (ALM) is presented. The scheme exploits the structure of the
Augmented Lagrangian Hessian. It is a modular preconditioner consisting of two
blocks. The first one is associated with the Lagrangian of the objective while
the second administers the Jacobian of the constraints and possible low-rank
corrections to the Hessian. The proposed updating strategies take advantage of
ALM convergence results and avoid frequent refreshing. Constraint
administration takes into account complementarity over the Lagrange multipliers
and admits relaxation. The preconditioner is designed for problems where
constraint quantity is small compared to the search space. A virtue of the
scheme is that it is agnostic to the preconditioning technique used for the
Hessian of the Lagrangian function. The strategy described can be used for
linear and nonlinear preconditioning. Numerical experiments report on spectral
properties of preconditioned matrices from Matrix Market while some
optimization problems where taken from the CUTEst collection. Preliminary
results indicate that the proposed scheme could be attractive and further
experimentation is encouraged.
