We focus on efficient preconditioning techniques for sequences of KKT linear systems
arising from the interior point solution of large convex quadratic programming problems.
Constraint Preconditioners~(CPs), though very effective in accelerating Krylov methods
in the solution of KKT systems, have a very high computational cost in some instances,
because their factorization
may be the most time-consuming task at each interior point iteration.
We overcome this problem by computing the CP from scratch only at selected interior point
iterations and by updating the last computed CP at the remaining iterations, via suitable
low-rank modifications based on a BFGS-like formula.
This work extends the limited-memory preconditioners for symmetric positive definite
matrices proposed by Gratton, Sartenaer and Tshimanga in [SIAM J. Optim. 2011; 21(3):912--935,
by exploiting specific features of KKT systems and CPs.
We prove that the updated preconditioners
still belong to the class of exact CPs, thus allowing the use of the conjugate gradient
method. Furthermore, they have the property of increasing the number of unit
eigenvalues of the preconditioned matrix as compared to generally used CPs.
Numerical experiments are reported, which show the effectiveness of our updating
technique when the cost for the factorization of the CP is high.
