Iterative solution of indefinite systems arising in large-scale optimization by the symmetric QMR algorithm
Many optimization algorithms, such as interior-point methods for linear and nonlinear programs or sequential programming methods for constrained nonlinear programs, require the solution of Kuhn-Tucker optimality conditions. Typically, this leads to linear systems with symmetric, but highly indefinite coefficient matrices. Often, these systems are very large and sparse and it is attractive to use iterative techniques for their solution. Unfortunately, existing algorithms for symmetric systems, such as SYMMLQ and MINRES, usually converge slowly for highly indefinite matrices. Furthermore, these schemes can be used only with positive definite preconditioners, which leaves the systems highly indefinite. In this talk, we propose a new iterative method for solving symmetric indefinite linear systems, which can be combined with general symmetric preconditioners. The algorithm can be interpreted as a special case of the QMR approach for non-Hermitian linear systems, which was recently proposed by Freund and Nachtigal, and, like the latter, it generates iterates defined by a quasi-minimal residual property. The proposed method has the same work and storage requirements per iteration as SYMMLQ or MINRES, however, it usually converges in considerably fewer iterations. We discuss the construction of efficient symmetric preconditioners for linear systems arising in optimization, and we report results of numerical experiments.
- OSTI ID:
- 36034
- Report Number(s):
- CONF-9408161--
- Country of Publication:
- United States
- Language:
- English
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