7 research outputs found
Some Preconditioning Techniques for Saddle Point Problems
Saddle point problems arise frequently in many applications in science and engineering, including constrained optimization, mixed finite element formulations of partial differential equations, circuit analysis, and so forth. Indeed the formulation of most problems with constraints gives rise to saddle point systems. This paper provides a concise overview of iterative approaches for the solution of such systems which are of particular importance in the context of large scale computation. In particular we describe some of the most useful preconditioning techniques for Krylov subspace solvers applied to saddle point problems, including block and constrained preconditioners.\ud
\ud
The work of Michele Benzi was supported in part by the National Science Foundation grant DMS-0511336
Updating constraint preconditioners for KKT systems in quadratic programming via low-rank corrections
This work focuses on the iterative solution of sequences of KKT linear
systems arising in interior point methods applied to large convex quadratic
programming problems. This task is the computational core of the interior point
procedure and an efficient preconditioning strategy is crucial for the
efficiency of the overall method. Constraint preconditioners are very effective
in this context; nevertheless, their computation may be very expensive for
large-scale problems, and resorting to approximations of them may be
convenient. Here we propose a procedure for building inexact constraint
preconditioners by updating a "seed" constraint preconditioner computed for a
KKT matrix at a previous interior point iteration. These updates are obtained
through low-rank corrections of the Schur complement of the (1,1) block of the
seed preconditioner. The updated preconditioners are analyzed both
theoretically and computationally. The results obtained show that our updating
procedure, coupled with an adaptive strategy for determining whether to
reinitialize or update the preconditioner, can enhance the performance of
interior point methods on large problems.Comment: 22 page
Recommended from our members
A comparative study of null-space factorizations for sparse symmetric saddle point systems
Null-space methods for solving saddle point systems of equations have long been used to transform an indefinite system into a symmetric positive definite one of smaller
dimension. A number of independent works in the literature have identified that we can interpret a null-space method as a matrix factorization. We review these findings, highlight links between them, and bring them into a unified framework.
We also investigate the suitability of using null-space factorizations to derive sparse direct methods, and present numerical results for both practical and academic problems
BFGS-like updates of constraint preconditioners for sequences of KKT linear systems in quadratic programming
We focus on efficient preconditioning techniques for sequences of KKT linear systems
arising from the interior point solution of large convex quadratic programming problems.
Constraint Preconditioners~(CPs), though very effective in accelerating Krylov methods
in the solution of KKT systems, have a very high computational cost in some instances,
because their factorization
may be the most time-consuming task at each interior point iteration.
We overcome this problem by computing the CP from scratch only at selected interior point
iterations and by updating the last computed CP at the remaining iterations, via suitable
low-rank modifications based on a BFGS-like formula.
This work extends the limited-memory preconditioners for symmetric positive definite
matrices proposed by Gratton, Sartenaer and Tshimanga in [SIAM J. Optim. 2011; 21(3):912--935,
by exploiting specific features of KKT systems and CPs.
We prove that the updated preconditioners
still belong to the class of exact CPs, thus allowing the use of the conjugate gradient
method. Furthermore, they have the property of increasing the number of unit
eigenvalues of the preconditioned matrix as compared to generally used CPs.
Numerical experiments are reported, which show the effectiveness of our updating
technique when the cost for the factorization of the CP is high
Recommended from our members
Approximate factorization constraint preconditioners for saddle-point matrices
We consider the application of the conjugate gradient method to the solution of large, symmetric indefinite linear systems. Special emphasis is put on the use of constraint preconditioners and a new factorization that can reduce the number of flops required by the preconditioning step. Results concerning the eigenvalues of the preconditioned matrix and its minimum polynomial are given. Numerical experiments validate these conclusions