  In this paper, we present a dynamic non-diagonal regularization for interior
point methods. The non-diagonal aspect of this regularization is implicit,
since all the off-diagonal elements of the regularization matrices are
cancelled out by those elements present in the Newton system, which do not
contribute important information in the computation of the Newton direction.
Such a regularization has multiple goals. The obvious one is to improve the
spectral properties of the Newton system solved at each iteration of the
interior point method. On the other hand, the regularization matrices introduce
sparsity to the aforementioned linear system, allowing for more efficient
factorizations. We also propose a rule for tuning the regularization
dynamically based on the properties of the problem, such that sufficiently
large eigenvalues of the non-regularized system are perturbed insignificantly.
This alleviates the need of finding specific regularization values through
experimentation, which is the most common approach in literature. We provide
perturbation bounds for the eigenvalues of the non-regularized system matrix
and then discuss the spectral properties of the regularized matrix. Finally, we
demonstrate the efficiency of the method applied to solve standard small and
medium-scale linear and convex quadratic programming test problems.
