Friday, May 4 2007, 11.00-12.00, Room 3721, Lindstedtsvägen 25

** Olaf Schenk**

University of Basel, Switzerland

E-mail: Olaf.Schenk@unibas.ch

We propose an inertia revealing preconditioning approach for the solution of

nonconvex constrained optimization problems. If interior methods with

second-derivative information are used for these optimization problems, a

linearized Karush-Kuhn-Tucker system of the optimality conditions has to be

solved. The main issue addressed is how to ensure that the Hessian is

positive definite in the null-space of the constraints while neither

adversely affecting the convergence of Newton's method or incurring a

significant computational overhead. In the nonconvex case, it is of

interest to find out the inertia of the current iteration system so that the

matrix may be modified a posteriori to obtain convergence to a minimum.

However, in order to not destroy the rapid convergence rate of the interior

method, the modification has only be performed in the cases where the

inertia is not correct and factorization methods [3, 4] are very often used

in order to compute the inertia information. In this work, we propose a new

inertia revealing preconditioned Krylov iteration to solve the linearized

Karush-Kuhn-Tucker system of optimality conditions. Our preconditioning

approaches for the symmetric indefinite Karush-Kuhn-Tucker systems are based

on maximum weighted matchings and algebraic multi-level incomplete $LBL^T$

factorizations [1, 2]. Finally, we present numerical results on several

large-scale three-dimensional examples of PDE-constrained optimizations in

the full space of states, control and adjoints variables with equality and

non-equality constraints and test them with artificial as well as clinical

data from biomedical cancer hyperthermia treatment planning.

[1] M. Hagemann and O. Schenk, Weighted matchings for the preconditioning of

symmetric indefinite linear systems, SIAM Journal of Scientific Computing,

28 (2006), pp. 403-420.

[2] O. Schenk, M.Bollhoefer, and R. Roemer, On large-scale diagonalization

techniques for the Anderson model of localization, SIAM Journal of

Scientific Computing, 28 (2006), pp. 963-983.

[3] O. Schenk and K. Gaertner, On fast factorization pivoting methods for

symmetric indefinite systems, Electronic Transaction of Numerical

Analysis, 23 (2006), pp. 158--179.

[4] O. Schenk, A. Waechter, and M. Hagemann, Matching-based preprocessing

algorithms to the solution of saddle-point problems in large-scale

nonconvex interior-point optimization, Journal of Computational

Optimization and Applications, Volume 36, Numbers 2-3 / April, 2007,

pp. 321-341

Calendar of seminars