Constraint-Preconditioned Krylov Solvers for Regularized Saddle-Point Systems

Abstract

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.