A Symmetric Reduction of the NT Direction

A stable symmetrization of the linear systems arising in interior-point methods for solving linear programs is introduced. A comparison of the condition numbers of the resulting interiorpoint linear systems with other commonly used approaches indicates that the new approach may be best suitable for an iterative solution. It is shown that there is a natural generalization of this symmetrization to the NT search direction for solving semidefinite programs. The generalization includes a novel pivoting strategy to minimize the norm of the right and side and heavily relies on the symmetry properties of the NT direction. As a byproduct, an approach to stabilize the systems of Schur-complement based interior-point solvers is derived. The search directions generated by iterative solvers typically have fairly low relative accuracy. Nevertheless, in some preliminary numerical examples, a suitably adapted interior point approach results in a rather small number of outer iterations.