  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(m) 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.
