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Civil-Comp Proceedings
ISSN 1759-3433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 89
Inexact Constraint Preconditioners for Optimization Problems L. Bergamaschi1, M. Venturin2 and G. Zilli1
1Department of Mathematical Methods and Models for Scientific Applications,
Full Bibliographic Reference for this paper
L. Bergamaschi, M. Venturin, G. Zilli, "Inexact Constraint Preconditioners for Optimization Problems", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", Civil-Comp Press, Stirlingshire, UK, Paper 89, 2006. doi:10.4203/ccp.84.89
Keywords: interior-point methods, iterative solvers, preconditioners, approximate Jacobian.
Summary
In this paper we are concerned with the solution of large scale minimization problems subject to equality constraints,
via interior point methods with iterative solvers.
When minimization problems subject
to equality constraints are considered, each iteration of an interior point
method requires the following linear system of equations
to be solved
where ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() There has been growing interest in recent years in the use of iterative methods to solve the linear system (77). Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system requiring less memory resources.
A variety of preconditioners have been proposed for such matrices.
They have a common feature of constructing
the approximation to (77) by simplifying its upper
left block, namely by applying the preconditioner of the form
where D is a sparse symmetric positive definite approximation to ![]() ![]() ![]()
The Hessian matrix
In this paper we look for a further simplification of the preconditioner:
we replace the Jacobian matrix A with a (sparse) approximation
We propose to use the preconditioner of the following form
where D is an approximation of ![]() ![]() ![]() ![]() ![]()
Dropping some of the elements in the Jacobian matrix A produces
a significant reduction of the fill-in of the Cholesky
factor of the preconditioner thus speeding-up the cost of a single
iteration of the Krylov subspace method of choice.
The spectral analysis of the preconditioned matrix reveals that
a large number of eigenvalues are one or positive and bounded
by those of References
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