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Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints∗

Published online by Cambridge University Press:  04 March 2013

Anton Schiela  and
Daniel Wachsmuth
Anton Schiela
Affiliation:
Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB), Takustraße 7, 14195 Berlin-Dahlem, Germany. schiela@zib.de; http://www.zib.de/schiela
Daniel Wachsmuth
Affiliation:
Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, A–4040 Linz, Austria; daniel.wachsmuth@oeaw.ac.at; http://www.ricam.oeaw.ac.at/people/page/wachsmuth
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Abstract

In the article an optimal control problem subject to a stationary variational inequality is investigated. The optimal control problem is complemented with pointwise control constraints. The convergence of a smoothing scheme is analyzed. There, the variational inequality is replaced by a semilinear elliptic equation. It is shown that solutions of the regularized optimal control problem converge to solutions of the original one. Passing to the limit in the optimality system of the regularized problem allows to prove C-stationarity of local solutions of the original problem. Moreover, convergence rates with respect to the regularization parameter for the error in the control are obtained, which turn out to be sharp. These rates coincide with rates obtained by numerical experiments, which are included in the paper.

Keywords

Variational inequalitiesoptimal controlcontrol constraintsregularizationC-stationaritypath-following
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 3 , May 2013 , pp. 771 - 787
Copyright
© EDP Sciences, SMAI, 2013

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References

Barbu, V., Optimal control of variational inequalities, Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), London 24 (1984). Google Scholar
Bergounioux, M. and Mignot, F., Optimal control of obstacle problems: existence of Lagrange multipliers. ESAIM: COCV 5 (2000) 45–70. Google Scholar
J.F. Bonnans and A. Shapiro, Perturbation analysis of optimization problems. Springer, New York (2000).
Brezis, H. and Stampacchia, G., Sur la régularité de la solution d’inéquations elliptiques. Bull. Soc. Math. France 96 (1968) 153–180. Google Scholar
Casas, E., Tröltzsch, F. and Unger, A., Second-order sufficient optimality conditions for a nonlinear elliptic control problem. J. Anal. Appl. 15 (1996) 687–707. Google Scholar
Dontchev, A.L., Implicit function theorems for generalized equations. Math. Program. A 70 (1995) 91–106. Google Scholar
A. Friedman, Variational principles and free-boundary problems. Pure and Applied Mathematics. John Wiley & Sons Inc., New York (1982).
Hintermüller, M. and Kopacka, I., Mathematical programs with complementarity constraints in function space: C- and strong stationarity and a path-following algorithm. SIAM J. Optim. 20 (2009) 868–902. Google Scholar
Hintermüller, M. and Kopacka, I., A smooth penalty approach and a nonlinear multigrid algorithm for elliptic MPECs. Comput. Optim. Appl. 50 (2011) 111–145. Google Scholar
M. Hintermüller, B.S. Mordukhovich and T. Surowiec, Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints. IFB-Report No. 46 (07/2011), Institute of Mathematics and Scientific Computing, University of Graz.
Hintermüller, M. and Surowiec, T., First-order optimality conditions for elliptic mathematical programs with equilibrium constraints via variational analysis. SIAM J. Optim. 21 (2011) 1561–1593. Google Scholar
L. Hörmander, The Analysis of Partial Differential Operators. Springer (1983).
Ito, K. and Kunisch, K., Optimal control of elliptic variational inequalities. Appl. Math. Optim. 41 (2000) 343–364. Google Scholar
Ito, K. and Kunisch, K., On the Lagrange multiplier approach to variational problems and applications, Monographs and Studies in Mathematics. SIAM, Philadelphia 24 (2008). Google Scholar
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980).
Kunisch, K. and Wachsmuth, D., Path-following for optimal control of stationary variational inequalities. Comp. Opt. Appl. 51 (2011) 1345–1373. Google Scholar
Kunisch, K. and Wachsmuth, D., Sufficient optimality conditions and semi-smooth Newton methods for optimal control of stationary variational inequalities. ESAIM: COCV 18 (2012). Google Scholar
Mignot, F., Contrôle dans les inéquations variationelles elliptiques. J. Funct. Anal. 22 (1976) 130–185. Google Scholar
Mignot, F. and Puel, J.-P., Optimal control in some variational inequalities. SIAM J. Control Optim. 22 (1984) 466–476. Google Scholar
Outrata, J., Jarušek, J. and Stará, J., On optimality conditions in control of elliptic variational inequalities. Set-Valued Var. Anal. 19 (2011) 23–42. Google Scholar
Scheel, H. and Scholtes, S., Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25 (2000) 1–22. Google Scholar
Stampacchia, G., Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189–258. Google Scholar
G. Wachsmuth, Private communication (2012).