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.
Classification : 49K20, 47J20, 49M15
Mots clés : variational inequalities, optimal control, sufficient optimality conditions, semi-smooth Newton method
@article{COCV_2012__18_2_520_0,
author = {Kunisch, Karl and Wachsmuth, Daniel},
title = {Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {520--547},
publisher = {EDP-Sciences},
volume = {18},
number = {2},
year = {2012},
doi = {10.1051/cocv/2011105},
zbl = {1246.49021},
mrnumber = {2954637},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2011105/}
}
TY - JOUR AU - Kunisch, Karl AU - Wachsmuth, Daniel TI - Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2012 DA - 2012/// SP - 520 EP - 547 VL - 18 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2011105/ UR - https://zbmath.org/?q=an%3A1246.49021 UR - https://www.ams.org/mathscinet-getitem?mr=2954637 UR - https://doi.org/10.1051/cocv/2011105 DO - 10.1051/cocv/2011105 LA - en ID - COCV_2012__18_2_520_0 ER -
Kunisch, Karl; Wachsmuth, Daniel. Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 2, pp. 520-547. doi : 10.1051/cocv/2011105. http://www.numdam.org/articles/10.1051/cocv/2011105/
[1] , Optimal control of variational inequalities, Monographs and Studies in Mathematics 24. Pitman, Advanced Publishing Program, London (1984). | MR 742624 | Zbl 0574.49005
[2] and , On the structure of Lagrange multipliers for state-constrained optimal control problems. Systems Control Lett. 48 (2003) 169-176. | MR 2020634 | Zbl 1134.49310
[3] and , Optimal control of obstacle problems : existence of Lagrange multipliers. ESAIM : COCV 5 (2000) 45-70. | Numdam | MR 1745686 | Zbl 0934.49008
[4] and , Sur la régularité de la solution d'inéquations elliptiques. Bull. Soc. Math. France 96 (1968) 153-180. | Numdam | MR 239302 | Zbl 0165.45601
[5] and , Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40 (2002) 1431-1454. | MR 1882801 | Zbl 1037.49024
[6] , and , Second order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations. SIAM J. Control Optim. 38 (2000) 1369-1391. | MR 1766420 | Zbl 0962.49016
[7] , and , Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19 (2008) 616-643. | MR 2425032 | Zbl 1161.49019
[8] , Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman, Advanced Publishing Program, Boston, MA (1985). | MR 775683 | Zbl 0695.35060
[9] 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 2515801 | Zbl 1189.49032
[10] and , Pde-constrained optimization subject to pointwise control and zero- or first-order state constraints. SIAM J. Optim. 17 (2006) 159-187. | Zbl 1137.49028
[11] and , An augmented Lagrangian technique for variational inequalities. Appl. Math. Optim. 21 (1990) 223-241. | MR 1036586 | Zbl 0692.49008
[12] and , Optimal control of elliptic variational inequalities. Appl. Math. Optim. 41 (2000) 343-364. | MR 1739400 | Zbl 0960.49003
[13] and , Semi-smooth Newton methods for variational inequalities of the first kind. ESAIM : M2AN 37 (2003) 41-62. | Numdam | MR 1972649 | Zbl 1027.49007
[14] and , On the Lagrange multiplier approach to variational problems and applications, Monographs and Studies in Mathematics 24. SIAM, Philadelphia (2008). | MR 2441683 | Zbl 1156.49002
[15] , Contrôle dans les inéquations variationelles elliptiques. J. Funct. Anal. 22 (1976) 130-185. | MR 423155 | Zbl 0364.49003
[16] and , Optimal control in some variational inequalities. SIAM J. Control Optim. 22 (1984) 466-476. | MR 739836 | Zbl 0561.49007
[17] and , Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dyn. Syst. 6 (2000) 431-450. | MR 1739375 | Zbl 1010.49015
[18] and , Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints. SIAM J. Optim. 17 (2006) 776-794. | MR 2257208 | Zbl 1119.49022
[19] and , Mathematical programs with complementarity constraints : stationarity, optimality, and sensitivity. Math. Oper. Res. 25 (2000) 1-22. | MR 1854317 | Zbl 1073.90557
[20] , Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus. Ann. Inst. Fourier (Grenoble) 15 (1965) 189-258. | Numdam | MR 192177 | Zbl 0151.15401
[21] , Optimale Steurung partieller Differentialgleichungen. Vieweg + Teubner, Wiesbaden (2009).
[22] and , Finitely additive measures. Trans. Am. Math. Soc. 72 (1952) 46-66. | MR 45194 | Zbl 0046.05401
Cité par Sources :
