SIAM Journal on Optimization, Volume 30, Issue 3, Page 1954-1979, January 2020.
We introduce a class of positive definite preconditioners for the solution of large symmetric indefinite linear systems or sequences of such systems, in optimization frameworks. The preconditioners are iteratively constructed by collecting information on a reduced eigenspace of the indefinite matrix by means of a Krylov-subspace solver. A spectral analysis of the preconditioned matrix shows the clustering of some eigenvalues and possibly the nonexpansion of its spectrum. Extensive numerical experimentation is carried out on standard difficult linear systems and by embedding the class of preconditioners within truncated Newton methods for large-scale unconstrained optimization (the issue of major interest). Although the Krylov-based method may provide modest information on matrix eigenspaces, the results obtained show that the proposed preconditioners lead to substantial improvements in terms of efficiency and robustness, particularly on very large nonconvex problems.

中文翻译：

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一类基于Krylov-子空间方法的近似逆预处理器用于大规模非凸优化

SIAM优化杂志，第30卷，第3期，第1954-1979页，2020年1月。
我们引入一类正定前置条件，用于在优化框架中求解大型对称不确定线性系统或此类系统的序列。通过使用Krylov子空间解算器收集关于不定矩阵的简化本征空间的信息来迭代构造前提条件。预处理矩阵的光谱分析显示了一些特征值的聚类，并且可能显示了其光谱的未扩展。在标准的困难线性系统上进行了广泛的数值实验，并将预处理器的类别嵌入到截断的牛顿方法中，以进行大规模的无约束优化（这是引起人们广泛关注的问题）。尽管基于Krylov的方法可能会提供有关矩阵特征空间的适度信息，