Augmented Lagrangian Techniques for Solving Saddle Point Linear Systems

We perform an algebraic analysis of a generalization of the augmented Lagrangian method for solution of saddle point linear systems. It is shown that in cases where the (1,1) block is singular, specifically semidefinite, a low-rank perturbation that minimizes the condition number of the perturbed matrix while maintaining sparsity is an effective approach. The vectors used for generating the perturbation are columns of the constraint matrix that form a small angle with the null-space of the original (1,1) block. Block preconditioning techniques of a similar flavor are also discussed and analyzed, and the theoretical observations are illustrated and validated by numerical results.