Preconditioners for indefinite systems arising in optimization
SIAM Journal on Matrix Analysis and Applications
Solving symmetric indefinite systems in an interior-point method for linear programming
Mathematical Programming: Series A and B
A QMR-based interior-point algorithm for solving linear programs
Mathematical Programming: Series A and B - Special issue: interior point methods in theory and practice
Primal-dual interior-point methods
Primal-dual interior-point methods
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
A Note on Preconditioning Nonsymmetric Matrices
SIAM Journal on Scientific Computing
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Preconditioning Indefinite Systems in Interior Point Methods for Optimization
Computational Optimization and Applications
A stable primal---dual approach for linear programming under nondegeneracy assumptions
Computational Optimization and Applications
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We propose to compute the search direction at each interior-point iteration for a linear program via a reduced augmented system that typically has a much smaller dimension than the original augmented system. This reduced system is potentially less susceptible to the ill-conditioning effect of the elements in the (1,1) block of the augmented matrix. A preconditioner is then designed by approximating the block structure of the inverse of the transformed matrix to further improve the spectral properties of the transformed system. The resulting preconditioned system is likely to become better conditioned toward the end of the interior-point algorithm. Capitalizing on the special spectral properties of the transformed matrix, we further proposed a two-phase iterative algorithm that starts by solving the normal equations with PCG in each IPM iteration, and then switches to solve the preconditioned reduced augmented system with symmetric quasi-minimal residual (SQMR) method when it is advantageous to do so. The experimental results have demonstrated that our proposed method is competitive with direct methods in solving large-scale LP problems and a set of highly degenerate LP problems.