Using constraint preconditioners with regularized saddle-point problems

Abstract

The problem of finding good preconditioners for the numerical solution of a certain important class of indefinite linear systems is considered. These systems are of a 2 by 2 block (KKT) structure in which the (2,2) block (denoted by −C) is assumed to be nonzero.In Constraint preconditioning for indefinite linear systems, SIAM J. Matrix Anal. Appl., 21 (2000), Keller, Gould and Wathen introduced the idea of using constraint preconditioners that have a specific 2 by 2 block structure for the case of C being zero. We shall give results concerning the spectrum and form of the eigenvectors when a preconditioner of the form considered by Keller, Gould and Wathen is used but the system we wish to solve may have C = 0. In particular, the results presented here indicate clustering of eigenvalues and, hence, faster convergence of Krylov subspace iterative methods when the entries of C are small; such a situations arise naturally in interior point methods for optimization and we present results for such problems which validate our conclusions.

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

“…We observe that systems (24), (25) and (20) are the first order optimality conditions of reduced quadratic optimization problems, obtained by eliminating all the constraints from the original quadratic problems. In other words, if condition (ii) holds, CP acts on the problem as a projector onto the space defined by the linear constraints.…”

“…An alternative approach is presented in [25], which consists in using CP with a zero (2,2)-block even if the KKT matrix may have D = 0. This idea is motivated by the observation that, when the IP method progresses toward the solution, the entries in D generally get smaller.…”

“…This may allow a significant reduction of the number of inner steps at the last outer IP iterations. As pointed out in [25], the use of such an approximate CP allows to perform its factorization once as long as J remains constant and M in (22) is chosen to remain constant too.…”

Usando grupos no cálculo do precondicionador separador aplicado aos métodos de pontos interiores

“…Isso motivou o desenvolvimento de precondicionadores para tais sistemas [1,2,6]. No entanto, com o sistema de equações normais,é possível utilizar o método dos gradientes conjugados, uma vez que a matriz de coeficientesé simétrica e definida positiva.…”

Sparse block factorization of saddle point matrices

“…Numerous solution methods for the saddle point systems of the form (1) can be found in the literature and many of them have focused on preconditioning techniques for Krylov subspace iterative solvers [1,2,3,7,12,18,23,24,26,28,31]. As a direct method against iterative solvers, various techniques on symmetric indefinite factorization P TÅ P = LDL T can be found in [9,14,21,34,35,38], where P is a permutation matrix, L is unit lower triangular matrix, D is block-diagonal matrix with blocks of order 1 or 2.…”