  We propose new preconditioned iterative solvers for linear systems arising in
primal-dual interior-point methods for convex quadratic programming problems.
These preconditioned conjugate gradient methods operate on an implicit Schur
complement of the KKT system at each iteration. In contrast to standard
approaches, the Schur complement we consider enables the reuse of the
factorization of the Hessian of the equality-constraint Lagrangian across all
interior point iterations. Further, the resulting reduced system admits
preconditioners that directly alleviate the ill-conditioning associated with
the strict complementarity condition in interior point methods. The two
preconditioners we propose also provably reduce the number of unique
eigenvalues for the coefficient matrix (CG iteration count). One is efficient
when the number of equality constraints is small, while the other is efficient
when the number of remaining degrees of freedom is small. Numerical experiments
with synthetic problems and problems from the Maros-M\'ezr\'aros QP collection
show that our preconditioned inexact interior point are effective at improving
conditioning and reducing cost. Across all test problems for which the direct
method is not fastest, our preconditioned methods achieve a reduction in cost
by a geometric mean of 1.432 relative to the best alternative preconditioned
method for each problem.
