SIAM Journal on Scientific Computing, Volume 43, Issue 2, Page A1001-A1026, January 2021.
We consider the iterative solution of regularized saddle-point systems. When the leading block is symmetric and positive semidefinite on an appropriate subspace, Dollar et al. [SIAM J. Matrix Anal. Appl., 28 (2006), pp. 170--189] describe how to apply the conjugate gradient (CG) method coupled with a constraint preconditioner, a choice that has proved to be effective in optimization applications. We investigate the design of constraint-preconditioned variants of other Krylov methods for regularized systems by focusing on the underlying basis-generation process. We build upon principles laid out by Gould, Orban, and Rees [SIAM J. Matrix Anal. Appl., 35 (2014), pp. 1329--1343] to provide general guidelines that allow us to specialize any Krylov method to regularized saddle-point systems. In particular, we obtain constraint-preconditioned variants of Lanczos and Arnoldi-based methods, including the Lanczos version of CG, MINRES, SYMMLQ, GMRES($\ell$), and DQGMRES. We also provide MATLAB implementations in hopes that they are useful as a basis for the development of more sophisticated software. Finally, we illustrate the numerical behavior of constraint-preconditioned Krylov solvers using symmetric and nonsymmetric systems arising from constrained optimization.

中文翻译：

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约束预处理的Krylov求解器，用于正则化鞍点系统

SIAM科学计算杂志，第43卷，第2期，第A1001-A1026页，2021年1月。
我们考虑正则化鞍点系统的迭代解。当前导块在适当的子空间上是对称且正半定的时，Dollar等人。[SIAM J.矩阵肛门。Appl。，28（2006），pp。170--189]描述了如何应用结合了约束条件预处理器的共轭梯度（CG）方法，这一选择已被证明在优化应用中是有效的。通过关注基础生成过程，我们研究了其他Krylov方法的正则化系统的约束预处理变体的设计。我们以Gould，Orban和Rees [SIAM J. Matrix Anal。Appl。，35（2014），pp。1329--1343]中提供一般性准则，使我们可以将任何Krylov方法专门用于正则化的鞍点系统。尤其是，我们获得基于Lanczos和Arnoldi的方法的约束条件预处理的变体，包括CG，MINRES，SYMMLQ，GMRES（$ \ ell $）和DQGMRES的Lanczos版本。我们还提供了MATLAB实现，希望它们可以作为开发更复杂软件的基础。最后，我们通过约束优化说明了使用对称和非对称系统进行约束预处理的Krylov求解器的数值行为。