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Schenk, Olaf ; Wächter, Andreas ; Weiser, Martin

Inertia Revealing Preconditioning For Large-Scale Nonconvex Constrained Optimization

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Kurzfassung in Englisch

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.

Freie Schlagwörter (englisch): nonconvex constrained optimization, interior-point method, inertia, multilevel incomplete factorization
MSC - Klassifikation 35B37
MSC - Klassifikation 49M15
MSC - Klassifikation 65F10
MSC - Klassifikation 65K10
Abteilung: Numerische Analysis und Modellierung
DDC-Sachgruppe: Mathematik
Dokumentart: ZIB-Report
Schriftenreihe: ZIB-Report
Band Nummer: 07-32
ISBN: 1438-0064
Quelle: Appeared in: SIAM J. Scientific Computing 31 (2008) 939-960
Sprache: Englisch
Erstellungsjahr: 2007
Publikationsdatum: 08.11.2007

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