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Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities

Published online by Cambridge University Press:  22 July 2011

Karl Kunisch  and
Daniel Wachsmuth
Karl Kunisch
Affiliation:
Institute for Mathematics and Scientific Computing, Heinrichstraße 36, 8010 Graz, Austria. karl.kunisch@uni-graz.at
Daniel Wachsmuth
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, 4040 Linz, Austria; daniel.wachsmuth@oeaw.ac.at
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Abstract

In this paper sufficient second order optimality conditions for optimal control problems subject to stationary variational inequalities of obstacle type are derived. Since optimality conditions for such problems always involve measures as Lagrange multipliers, which impede the use of efficient Newton type methods, a family of regularized problems is introduced. Second order sufficient optimality conditions are derived for the regularized problems as well. It is further shown that these conditions are also sufficient for superlinear convergence of the semi-smooth Newton algorithm to be well-defined and superlinearly convergent when applied to the first order optimality system associated with the regularized problems.

Keywords

Variational inequalitiesoptimal controlsufficient optimality conditionssemi-smooth Newton method
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 2 , April 2012 , pp. 520 - 547
Copyright
© EDP Sciences, SMAI, 2011

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