Direct Multiple Shooting for Nonlinear Optimum Experimental Design

  • Dennis JankaEmail author
  • Stefan Körkel
  • Hans Georg Bock
Conference paper
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 9)


Optimum experimental design (OED) for parameter identification has become a key technique in the model validation process for dynamical systems. This paper deals with optimum experimental design for systems modelled by differential-algebraic equations. We show how to formulate OED as a nonstandard nonlinear optimal control problem. The direct multiple shooting method is a state of the art method for the solution of standard optimal control problems that leads to structured nonlinear programs. We present two possibilities how to adapt direct multiple shooting to OED by introducing additional variables and constraints. We highlight special structures in the constraint and objective derivatives whose evaluation is usually the bottleneck when solving dynamic optimization problems by multiple shooting. We have implemented a structure exploiting algorithm that takes all these structures into account. Two benchmark examples show the efficiency of the new algorithm.


Optimal Control Problem Multiple Shooting Parameter Estimation Problem Path Constraint Optimum Experimental Design 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    Albersmeyer, J.: Adjoint based algorithms and numerical methods for sensitivity generation and optimization of large scale dynamic systems. Ph.D. thesis, Ruprecht-Karls-Universität Heidelberg (2010)Google Scholar
  2. 2.
    Atkinson, A.C., Donev, A.: Optimum Experimental Designs. Oxford Statistical Sciences Series, vol. 8. Oxford University Press, Oxford (1992)Google Scholar
  3. 3.
    Bauer, I., Bock, H.G., Schlöder J.P.: DAESOL – a BDF-code for the numerical solution of differential algebraic equations. Internal Report, IWR, SFB 359, Universität Heidelberg (1999)Google Scholar
  4. 4.
    Bauer, I., Bock, H.G., Körkel, S., Schlöder, J.P.: Numerical methods for optimum experimental design in DAE systems. J. Comput. Appl. Math. 120(1–2), 1–15 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bock, H.G.: Numerical treatment of inverse problems in chemical reaction kinetics. In: Ebert, K.H., Deuflhard, P., Jäger, W. (eds.) Modelling of Chemical Reaction Systems. Springer Series in Chemical Physics, vol. 18, pp. 102–125. Springer, Heidelberg (1981)CrossRefGoogle Scholar
  6. 6.
    Bock, H.G.: Randwertproblemmethoden zur parameteridentifizierung in systemen nichtlinearer differentialgleichungen. Bonner Mathematische Schriften, vol. 183. Universität Bonn, Bonn (1987)Google Scholar
  7. 7.
    Bock, H.G., Plitt, K.J.: A Multiple Shooting algorithm for direct solution of optimal control problems. In: Proceedings of the 9th IFAC World Congress, pp. 242–247. Pergamon Press, Budapest (1984). Available at
  8. 8.
    Bock, H.G. Eich, E., Schlöder. J.P.: Numerical solution of constrained least squares boundary value problems in differential-algebraic equations. In: Strehmel, K. (ed.) Numerical Treatment of Differential Equations. Proceedings of the NUMDIFF-4 Conference, Halle-Wittenberg, 1987. Texte zur Mathematik, vol. 104, pp. 269–280. Teubner, Leipzig (1988)Google Scholar
  9. 9.
    Contreras, M., Tapia, R.A.: Sizing the BFGS and DFP updates: numerical study. J. Optim. Theory Appl. 78(1), 93–108 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fedorov, V.V.: Theory of Optimal Experiments. Elsevier, Amsterdam (1972)Google Scholar
  11. 11.
    Ferreau, H.J., Kirches, C., Potschka, A., Bock, H.G., Diehl, M.: qpOASES: A parametric active-set algorithm for quadratic programming. Math. Program. Comput. 6(4), 327–363 (2014)Google Scholar
  12. 12.
    Franceschini, G., Macchietto, S.: Model-based design of experiments for parameter precision: state of the art. Chem. Eng. Sci. 63, 4846–4872 (2008)CrossRefGoogle Scholar
  13. 13.
    Hoang, M.D., Barz, T., Merchan, V.A., Biegler, L.T., Arellano-Garcia, H.: Simultaneous solution approach to model-based experimental design. AIChE J. 59(11), 4169–4183 (2013)CrossRefGoogle Scholar
  14. 14.
    Janka, D.: Optimum experimental design and multiple shooting. Master’s thesis, Universität Heidelberg, Heidelberg (2010)Google Scholar
  15. 15.
    Körkel, S.: Numerische methoden für optimale versuchsplanungsprobleme bei nichtlinearen DAE-modellen. Ph.D. thesis, Universität Heidelberg, Heidelberg (2002)Google Scholar
  16. 16.
    Körkel, S., Potschka, A., Bock, H.G., Sager, S.: A multiple shooting formulation for optimum experimental design. Math. Program. (2012, submitted revisions)Google Scholar
  17. 17.
    Potschka, A., Bock, H.G., Schlöder, J.P.: A minima tracking variant of semi-infinite programming for the treatment of path constraints within direct solution of optimal control problems. Optim. Methods Softw. 24(2), 237–252 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pukelsheim, F.: Optimal design of experiments. In: Classics in Applied Mathematics, vol. 50. SIAM, Philadelphia (2006). ISBN:978-0-898716-04-7.Google Scholar
  19. 19.
    Sager, S.: MIOCP benchmark site. (2014)
  20. 20.
    Sager, S.: Sampling decisions in optimum experimental design in the light of Pontryagin’s maximum principle. SIAM J. Control. Optim. 51(4), 3181–3207 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Wächter, A., Biegler, L.T.: Line search filter methods for nonlinear programming: motivation and global convergence. SIAM J. Optim. 16(1), 1–31 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Dennis Janka
    • 1
    Email author
  • Stefan Körkel
    • 1
  • Hans Georg Bock
    • 1
  1. 1.Interdisciplinary Center for Scientific ComputingHeidelberg UniversityHeidelbergGermany

Personalised recommendations