A new modified Cholesky factorization
SIAM Journal on Scientific and Statistical Computing
A taxonomy for conjugate gradient methods
SIAM Journal on Numerical Analysis
Changing the norm in conjugate gradient type algorithms
SIAM Journal on Numerical Analysis
Matrix computations (3rd ed.)
Primal-dual interior-point methods
Primal-dual interior-point methods
Block-Triangular Preconditioners for Saddle Point Problems with a Penalty Term
SIAM Journal on Scientific Computing
A Modified Cholesky Algorithm Based on a Symmetric Indefinite Factorization
SIAM Journal on Matrix Analysis and Applications
Constraint Preconditioning for Indefinite Linear Systems
SIAM Journal on Matrix Analysis and Applications
A Note on Preconditioning for Indefinite Linear Systems
SIAM Journal on Scientific Computing
Krylov Subspace Methods for Saddle Point Problems with Indefinite Preconditioning
SIAM Journal on Matrix Analysis and Applications
A Revised Modified Cholesky Factorization Algorithm
SIAM Journal on Optimization
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
CUTEr and SifDec: A constrained and unconstrained testing environment, revisited
ACM Transactions on Mathematical Software (TOMS)
Block triangular preconditioners for symmetric saddle-point problems
Applied Numerical Mathematics - Numerical algorithms, parallelism and applications
Algorithm 866: IFISS, a Matlab toolbox for modelling incompressible flow
ACM Transactions on Mathematical Software (TOMS)
Iterative Solution of Augmented Systems Arising in Interior Methods
SIAM Journal on Optimization
SIAM Journal on Matrix Analysis and Applications
On nonsymmetric saddle point matrices that allow conjugate gradient iterations
Numerische Mathematik
Modified Cholesky algorithms: a catalog with new approaches
Mathematical Programming: Series A and B
Combination Preconditioning and the Bramble-Pasciak$^{+}$ Preconditioner
SIAM Journal on Matrix Analysis and Applications
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Saddle-point systems arise in many applications areas, in fact in any situation where an extremum principle arises with constraints. The Stokes problem describing slow viscous flow of an incompressible fluid is a classic example coming from PDEs and in the area of optimization such problems are ubiquitous. In this paper we present a framework into which many well-known methods for solving saddle-point systems fit. Based on this description we show how new approaches for the solution of saddle-point systems arising in optimization can be derived from the Bramble-Pasciak conjugate gradient approach widely used in PDEs and more recent generalizations thereof. In particular we derive a class of new solution methods based on the use of preconditioned conjugate gradients in nonstandard inner products and demonstrate how these can be understood through more standard machinery. We show connections to constraint preconditioning and give the results of numerical computations on a number of standard optimization test examples.