Updating constraint preconditioners for KKT systems in quadratic programming via low-rank corrections

TL;DR

This paper introduces a low-rank correction method to efficiently update constraint preconditioners for KKT systems in large-scale quadratic programming, improving interior point method performance.

Contribution

It proposes a novel low-rank update approach for inexact constraint preconditioners, reducing computational costs in large convex quadratic programming problems.

Findings

The updating procedure improves convergence speed.

Adaptive strategy effectively balances reinitialization and updating.

Numerical results demonstrate enhanced performance on large problems.

Abstract

This work focuses on the iterative solution of sequences of KKT linear systems arising in interior point methods applied to large convex quadratic programming problems. This task is the computational core of the interior point procedure and an efficient preconditioning strategy is crucial for the efficiency of the overall method. Constraint preconditioners are very effective in this context; nevertheless, their computation may be very expensive for large-scale problems, and resorting to approximations of them may be convenient. Here we propose a procedure for building inexact constraint preconditioners by updating a "seed" constraint preconditioner computed for a KKT matrix at a previous interior point iteration. These updates are obtained through low-rank corrections of the Schur complement of the (1,1) block of the seed preconditioner. The updated preconditioners are analyzed both theoretically and computationally. The results obtained show that our updating procedure, coupled with an adaptive strategy for determining whether to reinitialize or update the preconditioner, can enhance the performance of interior point methods on large problems.