We consider the iterative solution of regularized saddle-point systems.
When the leading block is symmetric and positive semi-definite on an appropriate subspace,
Dollar, Gould, Schilders, and Wathen (2006) 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 (2014)
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.
Published October 2019 , 26 pages
This cahier was revised in July 2020