The interior-point method solves large linear programming problems in a few iterations. Each iteration requires computing the solution to one or more linear systems. This constitutes the most expensive step of the method and reducing the time to solve these linear systems is a way of improving the method’s performance. Iterative methods such as the preconditioned conjugate gradient method can be used to solve them. Incomplete Cholesky factorization can be used as a preconditioner to the problem. However, breakdowns may occur during incomplete factorizations and corrections on the diagonal may be required. This correction is accomplished by adding a positive number to diagonal elements of the linear system matrix and the factorization of the new matrix is restarted. In this work, we propose modifications to controlled Cholesky factorization to avoid or decrease the number of refactorizations of diagonally modified matrices. Computational results show that the proposed techniques can reduce the time needed for solving linear programming problems by the interior-point method.

内点法可以在几次迭代中解决大型线性规划问题。每次迭代都需要计算一个或多个线性系统的解。这是该方法中最昂贵的步骤，减少求解这些线性系统的时间是提高方法性能的一种方式。可以使用迭代方法如预处理共轭梯度法来解决它们。不完全 Cholesky 分解可以用作问题的预处理器。但是，在不完整的因式分解过程中可能会出现故障，可能需要对对角线进行更正。这种修正是通过向线性系统矩阵的对角元素添加一个正数来完成的，并且新矩阵的因式分解重新开始。在这项工作中，我们建议对受控 Cholesky 分解进行修改，以避免或减少对角修改矩阵的重构次数。计算结果表明，所提出的技术可以减少用内点法求解线性规划问题所需的时间。