Skip to main content
SpringerLink
Log in

An interior point method in function space for the efficient solution of state constrained optimal control problems

  • Full Length Paper
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

We propose and analyze an interior point path-following method in function space for state constrained optimal control. Our emphasis is on proving convergence in function space and on constructing a practical path-following algorithm. In particular, the introduction of a pointwise damping step leads to a very efficient method, as verified by numerical experiments.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

  1. Adams, R.A.: Sobolev Spaces. Academic, New York (1975)

    MATH  Google Scholar 

  2. Amann, H.: Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. In: Schmeisser, H.J., Triebel, H. (eds.) Function spaces, differential operators and nonlinear analysis, pp. 9–126. Teubner, Stuttgart, Leipzig (1993)

    Google Scholar 

  3. Bastian, P., Blatt, M., Engwer, C., Dedner, A., Klöfkorn, R., Kuttanikkad, S., Ohlberger, M., Sander, O.: The distributed and unified numerics environment (DUNE). In: Proceedings of the 19th Symposium on Simulation Technique in Hannover, Sep 12–14 (2006)

  4. Deuflhard, P.: Newton methods for nonlinear problems, 2nd edition. Affine Invariance and Adaptive Algorithms, Vol. 35 of Series Computational Mathematics. Springer (2006)

  5. Goldberg, S.: Unbounded Linear Operators. Dover Inc., New York (1966)

    MATH  Google Scholar 

  6. Götschel, S., Weiser, M., Schiela, A.: Solving Optimal Control Problems with the Kaskade 7 Finite Element Toolbox. In: Dedner, A., Flemisch, B., Klöfkorn, R. (eds.) Advances in DUNE, pp. 101–112. Springer, Berlin (2012)

    Chapter  Google Scholar 

  7. Haller-Dintelmann, R., Rehberg, J., Meyer, C., Schiela, A.: Hölder continuity and optimal control for nonsmooth elliptic problems. Appl. Math. Optim. 60(3), 397–428 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hintermüller, M., Kunisch, K.: Feasible and non-interior path-following in constrained minimization with low multiplier regularity. SIAM J. Control Optim. 45(4), 1198–1221 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hinze, M., Schiela, A.: Discretization of interior point methods for state constrained elliptic optimal control problems: optimal error estimates and parameter adjustment. Comput. Optim. Appl. 48(3), 581–600 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Meyer, C., Tröltzsch, F., Rösch, A.: Optimal control problems of PDEs with regularized pointwise state constraints. Comput. Optim. Appl. 33, 206–228 (2006)

    Article  Google Scholar 

  11. Prüfert, U., Tröltzsch, F., Weiser, M.: The convergence of an interior point method for an elliptic control problem with mixed control-state constraints. Comput. Optim. Appl. 39(2), 183–218 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Schenk, O., Gärtner, K.: On fast factorization pivoting methods for sparse symmetric indefinite systems. Electron. Trans. Numer. Anal. 23, 158–179 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Schiela, A.: The control reduced interior point method—a function space oriented algorithmic approach. PhD thesis, Department of Mathematics and Computer Science, Free University of Berlin (2006)

  14. Schiela, A.: Convergence of the Control Reduced Interior Point Method for PDE Constrained Optimal Control with State Constraints. ZIB Report 06–16, Zuse Institute Berlin (2006)

  15. Schiela, A.: Barrier methods for optimal control problems with state constraints. SIAM J. Optim. 20(2), 1002–1031 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Schiela, A.: An extended mathematical framework for barrier methods in function space. In: Domain Decomposition Methods in Science and Engineering XVIII, Vol. 70 of Lecture Notes in Computational Science and Engineering, pp. 201–208. Springer, Berlin (2009)

  17. Schiela, A., Günther, A.: An interior point algorithm with inexact step computation in function space for state constrained optimal control. Numerische Mathematik 119(2), 373–407 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Weiser, M.: Function space complementarity methods for optimal control problems. PhD thesis, Department of Mathematics and Computer Science, Free University of Berlin (2001)

Download references

Acknowledgments

The author wishes to thank Dr. Martin Weiser for helpful discussions and the close cooperation during the development of the computational framework.

Author information

Authors and Affiliations

  1. Zuse Institute Berlin, Takustraße 7, 14195 , Dahlem, Berlin, Germany

    Anton Schiela

Authors
  1. Anton Schiela
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anton Schiela.

Additional information

This study was supported by the DFG Research Center Matheon “Mathematics for key technologies”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Schiela, A. An interior point method in function space for the efficient solution of state constrained optimal control problems. Math. Program. 138, 83–114 (2013). https://doi.org/10.1007/s10107-012-0595-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-012-0595-y

Keywords

Mathematics Subject Classification (2000)

Search

Navigation