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
Primal-dual interior-point methods
Primal-dual interior-point methods
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
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Hyper-Sparsity in the Revised Simplex Method and How to Exploit it
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
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We discuss the use of preconditioned conjugate gradients (CG) method for solving the reduced KKT systems arising in interior point algorithms for linear programming. The (indefinite) augmented system form of this linear system has a number of advantages, notably a higher degree of sparsity than the (positive definite) normal equations form. Therefore, we use the CG method to solve the augmented system and look for a suitable preconditioner. An explicit null space representation of linear constraints is constructed by using a nonsingular basis matrix identified from an estimate of the optimal partition in the linear program. This is achieved by means of recently developed efficient basis matrix factorisation techniques which exploit hyper-sparsity and are used in implementations of the revised simplex method. The approach has been implemented within the HOPDM interior point solver and applied to medium and large-scale problems from public domain test collections. Computational experience is encouraging.