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.
Mots-clés : variational inequalities, optimal control, control constraints, regularization, C-stationarity, path-following
@article{M2AN_2013__47_3_771_0,
author = {Schiela, Anton and Wachsmuth, Daniel},
title = {Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {771--787},
publisher = {EDP-Sciences},
volume = {47},
number = {3},
year = {2013},
doi = {10.1051/m2an/2012049},
mrnumber = {3056408},
zbl = {1266.65112},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2012049/}
}
TY - JOUR AU - Schiela, Anton AU - Wachsmuth, Daniel TI - Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 771 EP - 787 VL - 47 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012049/ DO - 10.1051/m2an/2012049 LA - en ID - M2AN_2013__47_3_771_0 ER -
%0 Journal Article %A Schiela, Anton %A Wachsmuth, Daniel %T Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2013 %P 771-787 %V 47 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2012049/ %R 10.1051/m2an/2012049 %G en %F M2AN_2013__47_3_771_0
Schiela, Anton; Wachsmuth, Daniel. Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 3, pp. 771-787. doi : 10.1051/m2an/2012049. http://www.numdam.org/articles/10.1051/m2an/2012049/
[1] , Optimal control of variational inequalities, Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), London 24 (1984). | MR | Zbl
[2] and , Optimal control of obstacle problems: existence of Lagrange multipliers. ESAIM: COCV 5 (2000) 45-70. | Numdam | MR | Zbl
[3] and , Perturbation analysis of optimization problems. Springer, New York (2000). | MR | Zbl
[4] and , Sur la régularité de la solution d'inéquations elliptiques. Bull. Soc. Math. France 96 (1968) 153-180. | Numdam | MR | Zbl
[5] , and , Second-order sufficient optimality conditions for a nonlinear elliptic control problem. J. Anal. Appl. 15 (1996) 687-707. | MR | Zbl
[6] , Implicit function theorems for generalized equations. Math. Program. A 70 (1995) 91-106. | MR | Zbl
[7] , Variational principles and free-boundary problems. Pure and Applied Mathematics. John Wiley & Sons Inc., New York (1982). | MR | Zbl
[8] and , Mathematical programs with complementarity constraints in function space: C- and strong stationarity and a path-following algorithm. SIAM J. Optim. 20 (2009) 868-902. | MR | Zbl
[9] and , A smooth penalty approach and a nonlinear multigrid algorithm for elliptic MPECs. Comput. Optim. Appl. 50 (2011) 111-145. | MR | Zbl
[10] , and , 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.
[11] and , First-order optimality conditions for elliptic mathematical programs with equilibrium constraints via variational analysis. SIAM J. Optim. 21 (2011) 1561-1593. | MR | Zbl
[12] , The Analysis of Partial Differential Operators. Springer (1983). | Zbl
[13] and , Optimal control of elliptic variational inequalities. Appl. Math. Optim. 41 (2000) 343-364. | MR | Zbl
[14] and , On the Lagrange multiplier approach to variational problems and applications, Monographs and Studies in Mathematics. SIAM, Philadelphia 24 (2008). | MR | Zbl
[15] and , An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980). | MR | Zbl
[16] and , Path-following for optimal control of stationary variational inequalities. Comp. Opt. Appl. 51 (2011) 1345-1373. | MR | Zbl
[17] and , Sufficient optimality conditions and semi-smooth Newton methods for optimal control of stationary variational inequalities. ESAIM: COCV 18 (2012). | Numdam | MR | Zbl
[18] , Contrôle dans les inéquations variationelles elliptiques. J. Funct. Anal. 22 (1976) 130-185. | MR | Zbl
[19] and , Optimal control in some variational inequalities. SIAM J. Control Optim. 22 (1984) 466-476. | MR | Zbl
[20] , and , On optimality conditions in control of elliptic variational inequalities. Set-Valued Var. Anal. 19 (2011) 23-42. | Zbl
[21] and , Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25 (2000) 1-22. | MR | Zbl
[22] , Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189-258. | Numdam | MR | Zbl
[23] , Private communication (2012).
Cité par Sources :
