Regularization techniques in interior point methods

Regularization techniques, i.e., modifications on the diagonal elements of the scaling matrix, are considered to be important methods in interior point implementations. So far, regularization in interior point methods has been described for linear programming problems, in which case the scaling matrix is diagonal. It was shown that by regularization, free variables can be handled in a numerically stable way by avoiding column splitting that makes the set of optimal solutions unbounded. Regularization also proved to be efficient for increasing the numerical stability of the computations during the solutions of ill-posed linear programming problems. In this paper, we study the factorization of the augmented system arising in interior point methods. In our investigation, we generalize the methods developed and used in linear programming to the case when the scaling matrix is positive semidefinite, but not diagonal. We show that regularization techniques may be applied beyond the linear programming case.