We propose new preconditioned iterative solvers for linear systems arising in primal-dual interior-point methods for convex quadratic programming problems. These preconditioned conjugate gradient methods operate on an implicit Schur complement of the KKT system at each iteration. In contrast to standard approaches, the Schur complement we consider enables the reuse of the factorization of the Hessian of the equality-constraint Lagrangian across all interior point iterations. Further, the resulting reduced system admits preconditioners that directly alleviate the ill-conditioning associated with the strict complementarity condition in interior point methods. The two preconditioners we propose also provably reduce the number of unique eigenvalues for the coefficient matrix (CG iteration count). One is efficient when the number of equality constraints is small, while the other is efficient when the number of remaining degrees of freedom is small. Numerical experiments with synthetic problems and problems from the Maros-M\'ezr\'aros QP collection show that our preconditioned inexact interior point are effective at improving conditioning and reducing cost. Across all test problems for which the direct method is not fastest, our preconditioned methods achieve a reduction in cost by a geometric mean of 1.432 relative to the best alternative preconditioned method for each problem.
翻译:我们提出新的线性系统具有先决条件的迭代求解器,用于对锥形二次线性编程问题采用原始双点内点内点方法,这些具有先决条件的梯度方法在每次迭代时以KKT系统隐含的Schur补充形式运作。与标准办法不同,我们认为Schur补充器能够使平等限制拉格朗格的赫西安人重新利用所有内部点内点内点内点内点内点内点内线系统的因子化。此外,由此而减少的系统承认了直接减轻与内点方法的严格互补条件有关的不适应性的先决条件。我们建议的两个先决条件还可明显地减少系数矩阵(CGExeration country)中独有的埃因值数量。当平等限制的数量很小时,另一个是有效的,而当剩余自由度小时,则可以重新利用。马罗斯-M\ezr\'aroaro QP收集的合成问题和问题中的内点内点内点内点能有效改进和降低成本。我们提出的两个先决条件也是可以避免成本降低成本的。一个。在每种相对先决条件上,所有检验方法的最先期的地质方法,其最先期内点内点内点内点内点内点内点内点内点内点内点内点内点内点内点内,通过一种最能实现最佳的试验问题不是最能达到最佳减少成本。