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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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References

V. Barbu, Optimal control of variational inequalities, Monographs and Studies in Mathematics 24. Pitman, Advanced Publishing Program, London (1984).
Bergounioux, M. and Kunisch, K., On the structure of Lagrange multipliers for state-constrained optimal control problems. Systems Control Lett. 48 (2003) 169–176. Google Scholar
Bergounioux, M. and Mignot, F., Optimal control of obstacle problems : existence of Lagrange multipliers. ESAIM : COCV 5 (2000) 45–70. Google Scholar
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. and Mateos, M., Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40 (2002) 1431–1454. Google Scholar
Casas, E., Tröltzsch, F. and Unger, A., Second order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations. SIAM J. Control Optim. 38 (2000) 1369–1391. Google Scholar
Casas, E., de los Reyes, J.C. and Tröltzsch, F., Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19 (2008) 616–643. Google Scholar
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman, Advanced Publishing Program, Boston, MA (1985).
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 Kunisch, K., Pde-constrained optimization subject to pointwise control and zero- or first-order state constraints. SIAM J. Optim. 17 (2006) 159–187. Google Scholar
Ito, K. and Kunisch, K., An augmented Lagrangian technique for variational inequalities. Appl. Math. Optim. 21 (1990) 223–241. Google Scholar
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., Semi-smooth Newton methods for variational inequalities of the first kind. ESAIM : M2AN 37 (2003) 41–62. Google Scholar
K. Ito and K. Kunisch, On the Lagrange multiplier approach to variational problems and applications, Monographs and Studies in Mathematics 24. SIAM, Philadelphia (2008).
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
Raymond, J.-P. and Tröltzsch, F., Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dyn. Syst. 6 (2000) 431–450. Google Scholar
Rösch, A. and Tröltzsch, F., Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints. SIAM J. Optim. 17 (2006) 776–794. 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
F. Tröltzsch, Optimale Steurung partieller Differentialgleichungen. Vieweg + Teubner, Wiesbaden (2009).
Yosida, K. and Hewitt, E., Finitely additive measures. Trans. Am. Math. Soc. 72 (1952) 46–66. Google Scholar