We address the problem of preconditioning sequences of regularized KKT systems, such as those arising in Interior Point methods for convex quadratic programming. In this case, Constraint Preconditioners (CPs) are very effective and widely used; however, when solving large-scale problems, the computational cost for their factorization may be high, and techniques for approximating them appear to be a valid alternative. Here, given the block LDL^T factorization of the CP associated with a seed KKT matrix of the sequence, we propose a technique for updating such a factorization and building inexact CPs for subsequent matrices of the sequence. Very recently, we have proposed an updating procedure by performing a low-rank correction of the Schur complement of the (1,1) block of the CP for the seed matrix. Now we focus on KKT sequences with nonzero (2,2) block and make a step further, enriching the low-rank correction of the Schur complement by a further cheap update, which takes into account information not included in the previous procedure and expressed as a diagonal modification of the low-rank correction. Theoretical results and numerical experiments show that the new strategy can be more effective than the procedure based on the low-rank modification alone.
Citation
Published in SIAM Journal on Optimization Volume 25, Issue 3, 2015, Pages 1787-1808, DOI 10.1137/130947155