In general, interior point methods are successful in solving large-scale linear programming problems. Their effectiveness is determined by how fast they calculate each solution of a linear system. When solving large-scale linear systems, iterative methods are useful options since they require slightly more computational memory in each iteration and preserve the sparsity pattern of the matrix system. In this context, a common choice is the conjugate gradient method using a preconditioning strategy. Choosing a single preconditioner that fits well during all iterations of the optimization method is not an easy task, because the distribution of the eigenvalues of the system matrix may vary significantly from the first to the last iterations. In order to simplify this task, one can use different preconditioners in different iterations, in a hybrid preconditioner strategy. In the case of symmetric positive definite systems, the Controlled Cholesky Factorization achieves excellent performance for the first interior point iterations, whereas the Splitting preconditioner is very useful in the last iterations. However, since we apply a hybrid approach combining both preconditioners, it is necessary to decide when using each preconditioner. This paper addresses the critical issue of choosing between the two preconditioners of the hybrid strategy by using the condition number of the matrix system. The Ritz values obtained from the conjugate gradient method provide an approximation to the eigenvalues, which offers an estimate of the condition number. The main contribution of this research is a new heuristic to switching the preconditioners based on the estimated condition number. Numerical results for large-scale problems show that our choice to change the preconditioners adds both speed and robustness to a hybrid approach that combines the Controlled Cholesky Factorization and the Splitting preconditioner.

通常，内点方法可以成功解决大规模线性规划问题。它们的有效性取决于它们计算线性系统的每个解的速度。在求解大规模线性系统时，迭代方法是有用的选择，因为它们在每次迭代中都需要更多的计算内存，并保留矩阵系统的稀疏模式。在这种情况下，通常的选择是使用预处理策略的共轭梯度法。选择一个在优化方法的所有迭代过程中都适合的预处理器并不是一件容易的事，因为系统矩阵特征值的分布在第一次迭代到最后一次迭代中可能会发生很大变化。为了简化这项工作，您可以在不同的迭代中使用不同的前置条件，在混合前置条件策略中。在对称正定系统的情况下，可控的Cholesky因式分解在第一个内部点迭代中实现了出色的性能，而Splitting预条件器在最后的迭代中非常有用。但是，由于我们采用了将两个前置条件组合在一起的混合方法，因此有必要确定何时使用每个前置条件。本文通过使用矩阵系统的条件编号来解决在混合策略的两个前置条件之间进行选择的关键问题。从共轭梯度法获得的Ritz值提供了特征值的近似值，该特征值提供了条件数的估计值。这项研究的主要贡献是一种新的启发式方法，可以根据估计的条件数来切换预处理器。