We consider the identification of a distributed parameter in an elliptic variational inequality. On the basis of an optimal control problem formulation, the application of a primal-dual penalization technique enables us to prove the existence of multipliers giving a first order characterization of the optimal solution. Concerning the parameter we consider different regularity 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 algorithm emphasize the feasibility of our approach.
Mots-clés : bilevel problem, complementarity function, inverse problem, optimal control, variational inequality
@article{M2AN_2001__35_1_129_0,
author = {Hinterm\"uller, Michael},
title = {Inverse coefficient problems for variational inequalities : optimality conditions and numerical realization},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {129--152},
publisher = {EDP-Sciences},
volume = {35},
number = {1},
year = {2001},
mrnumber = {1811984},
zbl = {0978.65054},
language = {en},
url = {http://archive.numdam.org/item/M2AN_2001__35_1_129_0/}
}
TY - JOUR AU - Hintermüller, Michael TI - Inverse coefficient problems for variational inequalities : optimality conditions and numerical realization JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2001 SP - 129 EP - 152 VL - 35 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/item/M2AN_2001__35_1_129_0/ LA - en ID - M2AN_2001__35_1_129_0 ER -
%0 Journal Article %A Hintermüller, Michael %T Inverse coefficient problems for variational inequalities : optimality conditions and numerical realization %J ESAIM: Modélisation mathématique et analyse numérique %D 2001 %P 129-152 %V 35 %N 1 %I EDP-Sciences %U http://archive.numdam.org/item/M2AN_2001__35_1_129_0/ %G en %F M2AN_2001__35_1_129_0
Hintermüller, Michael. Inverse coefficient problems for variational inequalities : optimality conditions and numerical realization. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 1, pp. 129-152. http://archive.numdam.org/item/M2AN_2001__35_1_129_0/
[1] , Optimal Control of Variational Inequalities. Res. Notes Math., Pitman, 100 (1984). | MR | Zbl
[2] and, Control by the coefficients in a variational inequality: the inverse elastohydrodynamic lubrication problem. Report no. 173, I.N.S.A. Lyon (1994).
[3] , and, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam (1978). | MR | Zbl
[4] , Optimal control problems governed by abstract elliptic variational inequalities with state constraints. SIAM J. Control Optim. 36 (1998) 273-289. | Zbl
[5] and, Optimal control problems governed by obstacle type variational inequalities: a dual regularization penalization approach. J. Convex Anal. 5 (1998) 329-351. | Zbl
[6] , and, Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37 (1999) 1176-1194. | Zbl
[7] and, Optimal control of obstacle problems: existence of Lagrange multipliers. ESAIM: COCV 5 (2000) 45-70. | Numdam | Zbl
[8] and, 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). | MR | Zbl
[9] and, 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). | MR | Zbl
[10] , On a problem of the theory of lubrication governed by a variational inequality. Appl. Math. Optim. 3 (1977) 227-242. | Zbl
[11] , Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983). | MR | Zbl
[12] and, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Series in Computational Mathematics, Prentice-Hall, Englewood Cliffs, New Jersey (1983). | MR | Zbl
[13] , and, 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). | MR | Zbl
[14] , and, A smoothing method for mathematical programs with equilibrium constraints. Math. Prog. 85 (1999) 107-134. | Zbl
[15] , A variational inequality associated with a lubrication problem, IMA Preprint Series, no. 530 (1989).
[16] , A quasi-variational inequality arising in elastohydrodynamics. SIAM J. Math. Anal. 21 (1990) 18-36. | Zbl
[17] and, On the injectivity and linearization of the coefficient-to-solution mapping for elliptic boundary value problems. J. Math. Anal. Appl. 188 (1994) 1040-1066. | Zbl
[18] and, Optimal control of elliptic variational inequalities. Appl. Math. Optim. 41 (2000) 343-364. | Zbl
[19] and, Optimality conditions for strongly monotone variational inequalities. Appl. Math. Optim. 27 (1993) 291-312. | Zbl
[20] , and, Mathematical Programs with Equilibrium Constraints. Cambridge University Press, New York (1996). | MR | Zbl
[21] and, 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). | MR | Zbl
[22] and, Optimal control in some variational inequalities. SIAM J. Control Optim. 22 (1984) 466-476. | Zbl
