  We propose a novel preconditioned inexact primal-dual interior point method
for constrained convex quadratic programming problems. The algorithm we
describe invokes the preconditioned conjugate gradient method on a new reduced
Schur complement KKT system, in implicit form. In contrast to standard
approaches, the Schur complement formulation we consider enables reuse of the
factorization of the KKT matrix with rows and columns corresponding to
inequality constraints excluded, across all interior point iterations. Further,
two new preconditioners are presented for the resulting reduced system, that
alleviate the ill-conditioning associated with slack variables in primal-dual
interior point methods. Each of the preconditioners we propose also provably
reduces the number of unique eigenvalues for the coefficient matrix, and thus
the CG iteration count. One preconditioner 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\'esz\'aros QP collection show that our
preconditioned inexact interior point solvers 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.
