Recycling basic columns of the splitting preconditioner in interior point methods

Theoretical results and numerical experiments show that the linear systems originating from the last iterations of interior point methods (IPM) are very ill-conditioned. For this reason, preconditioners are necessary to approach this problem. In addition to that, in large-scale problems, the use of iterative methods and implicit preconditioners is essential because we only compute matrix–vector multiplications. Preconditioners with a lower computational cost than the splitting preconditioner only have good performance in the initial iterations of the IPM, so this preconditioner has become very important in the last iterations. The study of improvements thereof is justified. This paper studies the variation of the diagonal matrix D entries that appear in the linear systems to be solved to try to reuse or recycle some linearly independent columns of the splitting preconditioner base previously computed in a given IPM iteration to build another basis in the next one. It is justified by the fact that a subset of linearly independent columns remains linearly independent, and from that available subset, one may complete the number of columns necessary to form the new base. The numerical results show that the column recycling proposal improves the speed and robustness of the original approach for a test set, especially for large-scale problems.