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.
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}, mrnumber = {2954637}, zbl = {1246.49021}, 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 SP - 520 EP - 547 VL - 18 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2011105/ DO - 10.1051/cocv/2011105 LA - en ID - COCV_2012__18_2_520_0 ER -
%0 Journal Article %A Kunisch, Karl %A Wachsmuth, Daniel %T Sufficient optimality conditions and semi-smooth newton methods for optimal control of stationary variational inequalities %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 520-547 %V 18 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2011105/ %R 10.1051/cocv/2011105 %G en %F COCV_2012__18_2_520_0
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 | Zbl
,[2] On the structure of Lagrange multipliers for state-constrained optimal control problems. Systems Control Lett. 48 (2003) 169-176. | MR | Zbl
and ,[3] Optimal control of obstacle problems : existence of Lagrange multipliers. ESAIM : COCV 5 (2000) 45-70. | Numdam | MR | Zbl
and ,[4] Sur la régularité de la solution d'inéquations elliptiques. Bull. Soc. Math. France 96 (1968) 153-180. | Numdam | MR | Zbl
and ,[5] Second order optimality conditions for semilinear elliptic control problems with finitely many state constraints. SIAM J. Control Optim. 40 (2002) 1431-1454. | MR | Zbl
and ,[6] Second order sufficient optimality conditions for some state-constrained control problems of semilinear elliptic equations. SIAM J. Control Optim. 38 (2000) 1369-1391. | MR | Zbl
, and ,[7] Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19 (2008) 616-643. | MR | Zbl
, and ,[8] Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman, Advanced Publishing Program, Boston, MA (1985). | MR | Zbl
,[9] 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
and ,[10] Pde-constrained optimization subject to pointwise control and zero- or first-order state constraints. SIAM J. Optim. 17 (2006) 159-187. | Zbl
and ,[11] An augmented Lagrangian technique for variational inequalities. Appl. Math. Optim. 21 (1990) 223-241. | MR | Zbl
and ,[12] Optimal control of elliptic variational inequalities. Appl. Math. Optim. 41 (2000) 343-364. | MR | Zbl
and ,[13] Semi-smooth Newton methods for variational inequalities of the first kind. ESAIM : M2AN 37 (2003) 41-62. | Numdam | MR | Zbl
and ,[14] On the Lagrange multiplier approach to variational problems and applications, Monographs and Studies in Mathematics 24. SIAM, Philadelphia (2008). | MR | Zbl
and ,[15] Contrôle dans les inéquations variationelles elliptiques. J. Funct. Anal. 22 (1976) 130-185. | MR | Zbl
,[16] Optimal control in some variational inequalities. SIAM J. Control Optim. 22 (1984) 466-476. | MR | Zbl
and ,[17] Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete Contin. Dyn. Syst. 6 (2000) 431-450. | MR | Zbl
and ,[18] Sufficient second-order optimality conditions for an elliptic optimal control problem with pointwise control-state constraints. SIAM J. Optim. 17 (2006) 776-794. | MR | Zbl
and ,[19] Mathematical programs with complementarity constraints : stationarity, optimality, and sensitivity. Math. Oper. Res. 25 (2000) 1-22. | MR | Zbl
and ,[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 | Zbl
,[21] Optimale Steurung partieller Differentialgleichungen. Vieweg + Teubner, Wiesbaden (2009).
,[22] Finitely additive measures. Trans. Am. Math. Soc. 72 (1952) 46-66. | MR | Zbl
and ,Cité par Sources :