A sensitivity-based extrapolation technique for the numerical solution of state-constrained optimal control problems

Michael Hintermüller; Irwin Yousept

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 16, Issue: 3, page 503-522
  • ISSN: 1292-8119

Abstract

top
Sensitivity analysis (with respect to the regularization parameter) of the solution of a class of regularized state constrained optimal control problems is performed. The theoretical results are then used to establish an extrapolation-based numerical scheme for solving the regularized problem for vanishing regularization parameter. In this context, the extrapolation technique provides excellent initializations along the sequence of reducing regularization parameters. Finally, the favorable numerical behavior of the new method is demonstrated and a comparison to classical continuation methods is provided.

How to cite

top

Hintermüller, Michael, and Yousept, Irwin. "A sensitivity-based extrapolation technique for the numerical solution of state-constrained optimal control problems." ESAIM: Control, Optimisation and Calculus of Variations 16.3 (2010): 503-522. <http://eudml.org/doc/250740>.

@article{Hintermüller2010,
abstract = { Sensitivity analysis (with respect to the regularization parameter) of the solution of a class of regularized state constrained optimal control problems is performed. The theoretical results are then used to establish an extrapolation-based numerical scheme for solving the regularized problem for vanishing regularization parameter. In this context, the extrapolation technique provides excellent initializations along the sequence of reducing regularization parameters. Finally, the favorable numerical behavior of the new method is demonstrated and a comparison to classical continuation methods is provided. },
author = {Hintermüller, Michael, Yousept, Irwin},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Extrapolation; mixed control-state constraints; PDE-constrained optimization; semismooth Newton algorithm; sensitivity; state constraints; extrapolation; semismooth Newton algorithm},
language = {eng},
month = {7},
number = {3},
pages = {503-522},
publisher = {EDP Sciences},
title = {A sensitivity-based extrapolation technique for the numerical solution of state-constrained optimal control problems},
url = {http://eudml.org/doc/250740},
volume = {16},
year = {2010},
}

TY - JOUR
AU - Hintermüller, Michael
AU - Yousept, Irwin
TI - A sensitivity-based extrapolation technique for the numerical solution of state-constrained optimal control problems
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/7//
PB - EDP Sciences
VL - 16
IS - 3
SP - 503
EP - 522
AB - Sensitivity analysis (with respect to the regularization parameter) of the solution of a class of regularized state constrained optimal control problems is performed. The theoretical results are then used to establish an extrapolation-based numerical scheme for solving the regularized problem for vanishing regularization parameter. In this context, the extrapolation technique provides excellent initializations along the sequence of reducing regularization parameters. Finally, the favorable numerical behavior of the new method is demonstrated and a comparison to classical continuation methods is provided.
LA - eng
KW - Extrapolation; mixed control-state constraints; PDE-constrained optimization; semismooth Newton algorithm; sensitivity; state constraints; extrapolation; semismooth Newton algorithm
UR - http://eudml.org/doc/250740
ER -

References

top
  1. J.-J. Alibert and J.-P. Raymond, Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls. Numer. Funct. Anal. Optim.3/4 (1997) 235–250.  Zbl0885.49010
  2. E. Casas, Control of an elliptic problem with pointwise state contraints. SIAM J. Contr. Opt.4 (1986) 1309–1322.  Zbl0606.49017
  3. P. Deuflhard, Newton Methods for Nonlinear Problems. Affine Invariance and Adaptive Algorithms, Springer Series in Computational Mathematics35. Springer-Verlag, Berlin (2004).  Zbl1056.65051
  4. M. Hintermüller, Mesh-independence and fast local convergence of a primal-dual activ e–set method for mixed control-state constrained elliptic control problems. ANZIAM Journal49 (2007) 1–38.  Zbl1154.65057
  5. M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim.13 (2003) 865–888.  Zbl1080.90074
  6. M. Hintermüller and K. Kunisch, Feasible and non-interior path-following in constrained minimization with low multiplier regularity. SIAM J. Control Optim.45 (2006) 1198–1221.  Zbl1121.49030
  7. M. Hintermüller and K. Kunisch, Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim.17 (2006) 159–187.  Zbl1137.49028
  8. M. Hintermüller, F. Tröltzsch and I. Yousept, Mesh-independence of semismooth Newton methods for Lavrentiev-regularized state constrained nonlinear optimal control problems. Numer. Math.108 (2008) 571–603.  Zbl1143.65051
  9. M. Hinze and C. Meyer, Variational discretization of Lavrentiev-regularized state constrained elliptic optimal control problems. Computat. Optim. Appl. (2009), doi: .  Zbl1207.49037DOI10.1007/s10589-008-9198-1
  10. C. Meyer, A. Rösch and F. Tröltzsch, Optimal control of PDEs with regularized pointwise state constraints. Comp. Optim. Appl.33 (2006) 209–228.  Zbl1103.90072
  11. C. Meyer, U. Prüfert and F. Tröltzsch, On two numerical methods for stat e–constrained elliptic control problems. Optim. Method. Softw.22 (2007) 871–899.  Zbl1172.49022
  12. F. Tröltzsch, Regular Lagrange multipliers for control problems with mixed pointwise control-state constraints. SIAM J. Optim.15 (2004/2005) 616–634 (electronic).  Zbl1083.49018
  13. F. Tröltzsch and I. Yousept, A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints. Comp. Optim. Control42 (2009) 43–63.  Zbl1153.65343
  14. I. Yousept, Vergleich von Lösungsverfahren zur Behandlung elliptischer Optimalsteuerungsprobleme. Master's thesis, TU-Berlin, Germany (2005).  

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.