GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Direct methods for sparse matrices
Direct methods for sparse matrices
Preconditioners for indefinite systems arising in optimization
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Scientific and Statistical Computing
Software for simplified Lanczos and QMR algorithms
Applied Numerical Mathematics - Special issue on iterative methods for linear equations
Primal-dual interior-point methods
Primal-dual interior-point methods
An Iteration for Indefinite Systems and Its Application to the Navier--Stokes Equations
SIAM Journal on Scientific Computing
Journal of Optimization Theory and Applications
Constraint Preconditioning for Indefinite Linear Systems
SIAM Journal on Matrix Analysis and Applications
On the Solution of Equality Constrained Quadratic Programming Problems Arising in Optimization
SIAM Journal on Scientific Computing
Krylov Subspace Methods for Saddle Point Problems with Indefinite Preconditioning
SIAM Journal on Matrix Analysis and Applications
A Specialized Interior-Point Algorithm for Multicommodity Network Flows
SIAM Journal on Optimization
A Study of Preconditioners for Network Interior Point Methods
Computational Optimization and Applications
An interior-point approach for primal block-angular problems
Computational Optimization and Applications
Inexact constraint preconditioners for linear systems arising in interior point methods
Computational Optimization and Applications
Stopping criteria for inner iterations in inexact potential reduction methods: a computational study
Computational Optimization and Applications
Computational Optimization and Applications
Using constraint preconditioners with regularized saddle-point problems
Computational Optimization and Applications
Computational Optimization and Applications
Computational Optimization and Applications
Inner solvers for interior point methods for large scale nonlinear programming
Computational Optimization and Applications
Some iterative methods for the solution of a symmetric indefinite KKT system
Computational Optimization and Applications
Computational Optimization and Applications
Enhancing the behavior of interior-point methods via identification of variables
Optimization Methods & Software
Mixed Constraint Preconditioners for the iterative solution of FE coupled consolidation equations
Journal of Computational Physics
A stable primal---dual approach for linear programming under nondegeneracy assumptions
Computational Optimization and Applications
Computational Optimization and Applications
Journal of Computational and Applied Mathematics
Parallel inexact constraint preconditioners for saddle point problems
Euro-Par'11 Proceedings of the 17th international conference on Parallel processing - Volume Part II
The semi-convergence of generalized SSOR method for singular augmented systems
HPCA'09 Proceedings of the Second international conference on High Performance Computing and Applications
Matrix-free interior point method
Computational Optimization and Applications
Low-rank update of preconditioners for the inexact Newton method with SPD Jacobian
Mathematical and Computer Modelling: An International Journal
A preconditioning technique for Schur complement systems arising in stochastic optimization
Computational Optimization and Applications
GPU acceleration of the matrix-free interior point method
PPAM'11 Proceedings of the 9th international conference on Parallel Processing and Applied Mathematics - Volume Part I
Hi-index | 0.01 |
Every Newton step in an interior-point method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today's codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable ill-conditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used.