Efficient Preconditioners for Interior Point Methods via a new Schur Complementation Strategy

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\'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.