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Inverse Coefficient Problems for Variational Inequalities: Optimality Conditions and Numerical Realization

Published online by Cambridge University Press:  15 April 2002

Michael Hintermüller
Michael Hintermüller*
Affiliation:
Karl-Franzens University of Graz, Department of Mathematics, Heinrichstraße 36, 8010 Graz, Austria. ([email protected])
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Abstract

We consider the identification of a distributed parameter in an ellipticvariational inequality. On the basis of an optimal control problemformulation, the application of a primal-dual penalizationtechnique enables us to prove the existenceof multipliers giving a first order characterization of the optimal solution.Concerning the parameter we consider differentregularity requirements. For the numerical realization we utilize a complementarity function,which allows us to rewrite the optimality conditions as a set of equalities.Finally, numerical results obtained from a least squares type algorithmemphasize the feasibility of our approach.

Keywords

Bilevel problemcomplementarity functioninverse problemoptimal controlvariational inequality.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 1 , January 2001 , pp. 129 - 152
Copyright
© EDP Sciences, SMAI, 2001

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References

V. Barbu, Optimal Control of Variational Inequalities. Res. Notes Math., Pitman, 100 (1984).
B. Bayada and M. El Aalaoui Talibi, Control by the coefficients in a variational inequality: the inverse elastohydrodynamic lubrication problem. Report no. 173, I.N.S.A. Lyon (1994).
A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam (1978).
Bergounioux, M., Optimal control problems governed by abstract elliptic variational inequalities with state constraints. SIAM J. Control Optim. 36 (1998) 273-289. CrossRef
Bergounioux, M. and Dietrich, H., Optimal control problems governed by obstacle type variational inequalities: a dual regularization penalization approach. J. Convex Anal. 5 (1998) 329-351.
Bergounioux, M., Ito, K. and Kunisch, K., Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37 (1999) 1176-1194. CrossRef
Bergounioux, M. and Mignot, F., Optimal control of obstacle problems: existence of Lagrange multipliers. ESAIM: COCV 5 (2000) 45-70. CrossRef
A. Bermudez and C. Saguez, Optimality conditions for optimal control problems of variational inequalities, in: Control problems for systems described by partial differential equations and applications. I. Lasiecka and R. Triggiani Eds., Lect. Notes Control and Information Sciences, Springer, Berlin (1987).
G. Capriz and G. Cimatti, Free boundary problems in the theory of hydrodynamic lubrication: a survey, in: Free Boundary Problems: Theory and Applications, Vol. II, A. Fasano and M. Primicerio Eds., Res. Notes Math., Pitman, 79 (1983).
Cimatti, G., On a problem of the theory of lubrication governed by a variational inequality. Appl. Math. Optim. 3 (1977) 227-242. CrossRef
F. Clarke, Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983).
J. Dennis and R. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Series in Computational Mathematics, Prentice-Hall, Englewood Cliffs, New Jersey (1983).
F. Facchinei, A. Fischer and C. Kanzow, A semismooth Newton method for variational inequalities: the case of box constraints, in Complementarity and Variational Problems, State of the Art, M. Ferris and J. Pang Eds., SIAM, Philadelphia (1997).
Facchinei, F., Jiang, H. and Qi, L., A smoothing method for mathematical programs with equilibrium constraints. Math. Prog. 85 (1999) 107-134. CrossRef
J. Guo, A variational inequality associated with a lubrication problem, IMA Preprint Series, no. 530 (1989).
Hu, B., A quasi-variational inequality arising in elastohydrodynamics. SIAM J. Math. Anal. 21 (1990) 18-36. CrossRef
Ito, K. and Kunisch, K., On the injectivity and linearization of the coefficient-to-solution mapping for elliptic boundary value problems. J. Math. Anal. Appl. 188 (1994) 1040-1066. CrossRef
Ito, K. and Kunisch, K., Optimal control of elliptic variational inequalities. Appl. Math. Optim. 41 (2000) 343-364. CrossRef
Liu, W. and Rubio, J., Optimality conditions for strongly monotone variational inequalities. Appl. Math. Optim. 27 (1993) 291-312. CrossRef
Z. Luo, J. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints. Cambridge University Press, New York (1996).
Z. Luo and P. Tseng, A new class of merit functions for the nonlinear complementarity problem, in Complementarity and Variational Problems, State of the Art, M. Ferris and J. Pang Eds., SIAM, Philadelphia (1997).
Mignot, F. and Puel, J.P., Optimal control in some variational inequalities. SIAM J. Control Optim. 22 (1984) 466-476. CrossRef