### Sue Dollar (Rutherford Appleton Laboratory)

#### Iterative algebra for constrained optimization

*Joint work with Nick Gould, Wil Schilders and Andy Wathen.*

*Wednesday 15 November 2006 at 15.30, JCMB 5327*

##### Abstract

Each step of an interior point method for linear and non-linear optimization
requires the solution of a system of equations which is symmetric and
indefinite. Such a system is often called a saddle point or KKT problem. As the
problem size increases, the use of direct methods to solve this system may
become prohibitively expensive and iterative methods become a viable
alternative.

We will propose the use of implicitly defined constraint preconditioners
when (approximately) solving saddle point problems iteratively. These
preconditioners only require the factorization of smaller sparse systems
and can be shown to give favourable Krylov subspace properties as well
as allowing us to use a conjugate gradient-based iterative method.
However, they often require a preprocessing step which carries out a
symmetric permutation of the saddle point problem. This permutation
strongly relates to finding a nullspace basis of the linearized
constraint space for the underlying optimization problem, but we would
also like to take additional characteristics of the Hessian into account
when forming the permutation. We will discuss the desirable properties
of such a permutation and how this might be achieved. Numerical
experiments comparing several possible methods will be presented.

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