## A QMR-based interior-point algorithm for solving linear programs

### R.W. Freund and F. Jarre

A new approach for the implementation of interior-point methods for solving linear programs is proposed. Its main feature is the iterative solution of the symmetric, but highly indefinite $2 \times 2$-block systems of linear equations that arise within the interior-point algorithm. These linear systems are solved by a symmetric variant of the quasi-minimal residual (QMR) algorithm, which is an iterative solver for general linear systems. The symmetric QMR algorithm can be combined with indefinite preconditioners, which is crucial for the efficient solution of highly indefinite linear systems, yet it still fully exploits the symmetry of the linear systems to be solved. To support the use of the symmetric QMR iteration, a novel stable reduction of the original unsymmetric $3 \times 3$-block systems to symmetric $2 \times 2$-block systems is introduced, and a measure for a low relative accuracy for the solution of these linear systems within the interior-point algorithm is proposed. Some indefinite preconditioners are discussed. Finally, we report results of a few preliminary numerical experiments to illustrate the features of the new approach.

AT&T Numerical Analysis Manuscript No. 94-19, Bell Laboratories, Murray Hill, New Jersey, December 1994.