Constraint-Style Preconditioners for Regularized Saddle Point Problems

Abstract

The problem of finding good preconditioners for the numerical solution of an important class of indefinite linear systems is considered. These systems are of a regularized saddle point structure [ A B T B −C ][ x y ] = [ c d ], where A ∈ R n×n , C ∈ R m×m are symmetric and B ∈ R m×n. In [SIAM J. Matrix Anal. Appl., 21 (2000), pp. 1300-1317], Keller, Gould, and Wathen analyze the idea of using constraint preconditioners that have a specific 2 by 2 block structure for the case of C being zero. We shall extend this idea by allowing the (2, 2) block to be symmetric and positive semidefinite. Results concerning the spectrum and form of the eigenvectors are presented, as are numerical results to validate our conclusions.

On mutual impact of numerical linear algebra and large-scale optimization with focus on interior point methods

“…In view of Theorem 6, we observe that the better M approximates H, the more clustered around 1 are the eigenvalues of P −1 K. The results of the previous theorem can be extended to the case D positive semidefinite, as follows [23].…”

“…any method that generates an orthogonal basis for the corresponding Krylov subspace, terminates after at most n − m + 2 steps [54]. Such result has been extended to the case D positive semidefinite [23]. Indeed, if the assumptions of Theorem 7 hold and Z T (M + J T I D −1 I J I )Z is positive definite, then, in exact arithmetic, any optimal Krylov method terminates after at most min{n − m + p + 2, n + m} steps.…”

“…A CP variant where D instead of H is approximated has been analysed in [28]; the resulting preconditioner is generally more expensive to apply, unless optimization problems with a particular structure are considered. The spectral properties of the preconditioned matrix P −1 K have been investigated in many papers [6,8,23,28,31,54,57]. The following result holds for the case D = 0 [54].…”

“…CPs date back to the 70's [1], but in the last decade they have been extensively applied to the KKT system, not only in the optimization context (see e.g. [7,8,13,23,24,28,31,45,47,54,57,70]). We consider the following symmetric indefinite CP:…”

“…with preconditioner Z T (M + J T I D −1 I J I )Z [23]; finally, when D is nonsingular (i.e. positive definite), CG with CP is equivalent to applying CG to the condensed system (20) with preconditioner M + J T D −1 J.…”