Stability of Augmented System Factorizations in Interior-point Methods

Some implementations of interior-point algorithms obtain their search directions by solving symmetric indeenite systems of linear equations. The conditioning of the coeecient matrices in these so-called augmentedsystems deteriorateson later iterations, as some of the diagonal elements grow without bound. Despite this apparent diiculty, the steps produced by standard factorization procedures are often accurate enough to allow the interior-point method to converge to high accuracy. When the underlying linear program is nondegenerate, we show that convergence to arbitrarily high accuracy occurs, at a rate that closely approximates the theory. We also explain and demonstrate what happens when the linear program is degenerate, where convergence to acceptable accuracy (but not arbitrarily high accuracy) is usually obtained.