Inertia-controlling factorizations for optimization algorithms
Applied Numerical Mathematics
Inner solvers for interior point methods for large scale nonlinear programming
Computational Optimization and Applications
Matrix-free interior point method
Computational Optimization and Applications
Pivoting strategies for tough sparse indefinite systems
ACM Transactions on Mathematical Software (TOMS)
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Sparse linear equations $Kd=r$ are considered, where $K$ is a specially structured symmetric indefinite matrix that arises in numerical optimization and elsewhere. Under certain conditions, $K$ is quasidefinite. The Cholesky factorization $PKP^{T} = LDL^{T}$ is then known to exist for any permutation $P$, even though $D$ is indefinite. Quasidefinite matrices have been used successfully by Vanderbei within barrier methods for linear and quadratic programming. An advantage is that for a sequence of $K$'s, $P$ may be chosen once and for all to optimize the sparsity of $L$, as in the positive-definite case. A preliminary stability analysis is developed here. It is observed that a quasidefinite matrix is closely related to an unsymmetric positive-definite matrix, for which an $LDM^{T}$ factorization exists. Using the Golub and Van Loan analysis of the latter, conditions are derived under which Cholesky factorization is stable for quasidefinite systems. Some numerical results confirm the predictions.