For any indefinite matrix K, it is shown that the UTDU factorization can be modified at nominal cost to provide an "exact" preconditioner for SYMMLQ. Code is given for overwriting the blockdiagonal matrix D produced by MA27.The KKT systems arising in barrier methods for linear and nonlinear programming are studied, and preconditioners for use with SYMMLQ are derived.For nonlinear programs a preconditioner is derived from the "smaller" KKT system associated with variables that are not near a bound. For linear programs several preconditioners are proposed, based on a square nonsingular matrix B that is analogous to the basis matrix in the simplex method.The aim is to facilitate solution of full KKT systems rather than equations of the form AD2ATAr r when the latter become excessively ill conditioned.

In barrier methods for constrained optimization, the main work lies in solving large linear systems Kp = r, where K is symmetric and indefinite.For linear programs, these KKT systems are usually reduced to smaller positive-definite systems AH AT is typically more ill-conditioned than K.In order to improve the numerical properties of barrier implementations, we discuss the use of "reduced KKT systems", whose dimension and condition lie somewhere in between those of K and AH −1 A T . The approach applies to linear programs and to positive semidefinite quadratic programs whose Hessian H is at least partially diagonal.We have implemented reduced KKT systems in a primal-dual algorithm for linear programming, based on the sparse indefinite solver MA27 from the Harwell Subroutine Library. Some features of the algorithm are presented, along with results on the netlib LP test set.

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Many interior-point methods for linear programming are based on the properties of the logarithmic barrier function. We first give a convergence proof for the (primal) projected Newton barrier method. We then analyze three types of barrier method that can be categorized as primal, dual and primal-dual. All three approaches may be derived from the application of Newton's method to different variants of the same system of nonlinear equations. A fourth variant of the same equations leads to a new primal-dual algorithm.In each of the methods discussed, convergence is demonstrated without the need for a nondegeneracy assumption. In particular, convergence is established for a primal-dual algorithm that allows a different step in the primal and dual variables.Finally, we describe a new method for treating free variables.