1031
eng
reportzib
0
2007-11-08
2007-11-08
--
Inertia Revealing Preconditioning For Large-Scale Nonconvex Constrained Optimization
Fast nonlinear programming methods following the all-at-once approach usually employ Newton's method for solving linearized Karush-Kuhn-Tucker (KKT) systems. In nonconvex problems, the Newton direction is only guaranteed to be a descent direction if the Hessian of the Lagrange function is positive definite on the nullspace of the active constraints, otherwise some modifications to Newton's method are necessary. This condition can be verified using the signs of the KKT's eigenvalues (inertia), which are usually available from direct solvers for the arising linear saddle point problems. Iterative solvers are mandatory for very large-scale problems, but in general do not provide the inertia. Here we present a preconditioner based on a multilevel incomplete $LBL^T$ factorization, from which an approximation of the inertia can be obtained. The suitability of the heuristics for application in optimization methods is verified on an interior point method applied to the CUTE and COPS test problems, on large-scale 3D PDE-constrained optimal control problems, as well as 3D PDE-constrained optimization in biomedical cancer hyperthermia treatment planning. The efficiency of the preconditioner is demonstrated on convex and nonconvex problems with $150^3$ state variables and $150^2$ control variables, both subject to bound constraints.
07-32
1438-0064
1060
urn:nbn:de:0297-zib-10314
Appeared in: SIAM J. Scientific Computing 31 (2008) 939-960
Olaf Schenk
unknown unknown
Andreas Wächter
Martin Weiser
ZIB-Report
07-32
eng
uncontrolled
nonconvex constrained optimization
eng
uncontrolled
interior-point method
eng
uncontrolled
inertia
eng
uncontrolled
multilevel incomplete factorization
Mathematik
Newton-type methods
Iterative methods for linear systems [See also 65N22]
Optimization and variational techniques [See also 49Mxx, 93B40]
Numerical Mathematics
Computational Medicine
Weiser, Martin
MATHEON-A17
ZIB-Kaskade7
https://opus4.kobv.de/opus4-zib/files/1031/ZR_07_32.pdf