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

• Volume: 35, Issue: 1, page 129-152
• ISSN: 0764-583X

top

## Abstract

top
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.

## How to cite

top

Hintermüller, Michael. "Inverse Coefficient Problems for Variational Inequalities: Optimality Conditions and Numerical Realization." ESAIM: Mathematical Modelling and Numerical Analysis 35.1 (2010): 129-152. <http://eudml.org/doc/197552>.

@article{Hintermüller2010,
abstract = { 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. },
author = {Hintermüller, Michael},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Bilevel problem; complementarity function; inverse problem; optimal control; variational inequality.; elliptic variational inequality; inverse elastohydrodynamic lubrication problem; least squares method; primal-dual penalization technique; complementarity functions; algorithm; Gauss-Newton method; numerical tests},
language = {eng},
month = {3},
number = {1},
pages = {129-152},
publisher = {EDP Sciences},
title = {Inverse Coefficient Problems for Variational Inequalities: Optimality Conditions and Numerical Realization},
url = {http://eudml.org/doc/197552},
volume = {35},
year = {2010},
}

TY - JOUR
AU - Hintermüller, Michael
TI - Inverse Coefficient Problems for Variational Inequalities: Optimality Conditions and Numerical Realization
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 35
IS - 1
SP - 129
EP - 152
AB - 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.
LA - eng
KW - Bilevel problem; complementarity function; inverse problem; optimal control; variational inequality.; elliptic variational inequality; inverse elastohydrodynamic lubrication problem; least squares method; primal-dual penalization technique; complementarity functions; algorithm; Gauss-Newton method; numerical tests
UR - http://eudml.org/doc/197552
ER -

## References

top
1. V. Barbu, Optimal Control of Variational Inequalities. Res. Notes Math., Pitman, 100 (1984).
2. 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).
3. A. Bensoussan, J.L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam (1978).  Zbl0404.35001
4. M. Bergounioux, Optimal control problems governed by abstract elliptic variational inequalities with state constraints. SIAM J. Control Optim.36 (1998) 273-289.  Zbl0919.49002
5. M. Bergounioux and H. Dietrich, Optimal control problems governed by obstacle type variational inequalities: a dual regularization penalization approach. J. Convex Anal.5 (1998) 329-351.  Zbl0919.49003
6. M. Bergounioux, K. Ito and K. Kunisch, Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim.37 (1999) 1176-1194.  Zbl0937.49017
7. M. Bergounioux and F. Mignot, Optimal control of obstacle problems: existence of Lagrange multipliers. ESAIM: COCV5 (2000) 45-70.  Zbl0934.49008
8. 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).  Zbl0627.49011
9. 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).  Zbl0557.76038
10. G. Cimatti, On a problem of the theory of lubrication governed by a variational inequality. Appl. Math. Optim.3 (1977) 227-242.  Zbl0404.76036
11. F. Clarke, Optimization and Nonsmooth Analysis. Wiley-Interscience, New York (1983).  Zbl0582.49001
12. J. Dennis and R. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Series in Computational Mathematics, Prentice-Hall, Englewood Cliffs, New Jersey (1983).  Zbl0579.65058
13. 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).  Zbl0886.90152
14. F. Facchinei, H. Jiang and L. Qi, A smoothing method for mathematical programs with equilibrium constraints. Math. Prog.85 (1999) 107-134.  Zbl0959.65079
15. J. Guo, A variational inequality associated with a lubrication problem, IMA Preprint Series, no. 530 (1989).
16. B. Hu, A quasi-variational inequality arising in elastohydrodynamics. SIAM J. Math. Anal.21 (1990) 18-36.  Zbl0718.35101
17. K. Ito and K. Kunisch, On the injectivity and linearization of the coefficient-to-solution mapping for elliptic boundary value problems. J. Math. Anal. Appl.188 (1994) 1040-1066.  Zbl0817.35021
18. K. Ito and K. Kunisch, Optimal control of elliptic variational inequalities. Appl. Math. Optim.41 (2000) 343-364.  Zbl0960.49003
19. W. Liu and J. Rubio, Optimality conditions for strongly monotone variational inequalities. Appl. Math. Optim.27 (1993) 291-312.  Zbl0779.49012
20. Z. Luo, J. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints. Cambridge University Press, New York (1996).  Zbl1139.90003
21. 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).  Zbl0886.90158
22. F. Mignot and J.P. Puel, Optimal control in some variational inequalities. SIAM J. Control Optim.22 (1984) 466-476.  Zbl0561.49007

## NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.