On a Fully Adaptive SQP Method for PDAE-Constrained Optimal Control Problems with Control and State Constraints

  • Stefanie Bott
  • Debora Clever
  • Jens Lang
  • Stefan UlbrichEmail author
  • Jan Carsten Ziems
  • Dirk Schröder
Part of the International Series of Numerical Mathematics book series (ISNM, volume 165)


We present an adaptive multilevel optimization approach which is suitable to solve complex real-world optimal control problems for time-dependent nonlinear partial differential algebraic equations with point-wise constraints on control and state. Relying on Moreau-Yosida regularization, the multilevel SQP method presented in Clever et al. (Generalized multilevel SQP-methods for PDAE-constrained optimization based on space-time adaptive PDAE solvers. In: Constrained optimization and optimal control for partial differential equations. Volume 160 of International series of numerical mathematics. Springer, Basel, pp 37–60, 2012) is extended to the state-constrained case. First-order convergence results are shown. The new multilevel SQP method is combined with the state-of-the-art software package KARDOS to allow the efficient resolution of different space and time scales in an adaptive manner. The numerical performance of the method is demonstrated and analyzed for a real-life three-dimensional radiative heat transfer problem modeling the cooling process in glass manufacturing and a two-dimensional thermistor problem modeling the heating process in steel hardening.


PDAE-constrained optimization Multilevel optimization Generalized SQP method Trust region methods Control constraints State constraints Moreau-Yosida regularization Adaptive finite elements Rosenbrock methods Glass cooling Radiation Steel hardening 

Mathematics Subject Classification (2010)

49J20 49M25 65C20 65K10 65M60 90C46 90C55 



The authors gratefully acknowledge the support of the German Research Foundation (DFG) within the Priority Program 1253 “Optimization with Partial Differential Equations” under grants LA1372/6-2 and UL158/7-2 and the support by the ‘Excellence Initiative’ of the German Federal and State Governments within the Graduate School of Computational Engineering at Technische Universität Darmstadt.


  1. 1.
    D.P. Bertsekas, Projected Newton methods for optimization problems with simple constraints. SIAM J. Control Optim. 20, 221–246 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    D. Clever, Adaptive multilevel methods for PDAE-constrained optimal control problems. PhD thesis, Technische Universität Darmstadt, 2013. Verlag Dr. HutGoogle Scholar
  3. 3.
    D. Clever, J. Lang, Optimal control of radiative heat transfer in glass cooling with restrictions on the temperature gradient. Optim. Control Appl. Methods 33(2), 157–175 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    D. Clever, J. Lang, D. Schröder, Model hierarchy based optimal control of radiative heat transfer. Int. J. Comput. Sci. Eng. 9(5/6), 509–525 (2014)CrossRefGoogle Scholar
  5. 5.
    D. Clever, J. Lang, S. Ulbrich, J.C. Ziems, Generalized multilevel SQP-methods for PDAE-constrained optimization based on space-time adaptive PDAE solvers, in Constrained Optimization and Optimal Control for Partial Differential Equations. Volume 160 of International Series of Numerical Mathematics (Springer, Basel, 2012), pp. 37–60Google Scholar
  6. 6.
    K. Debrabant, J. Lang, On global error estimation and control of finite difference solution for parabolic equations, in Adaptive Modeling and Simulation (International Center for Numerical Methods in Engineering (CIMNE), Barcelona, 2013), pp. 187–198Google Scholar
  7. 7.
    B. Erdmann, J. Lang, R. Roitzsch, KARDOS-User’s Guide. Manual, Konrad-Zuse-Zentrum Berlin, 2002Google Scholar
  8. 8.
    M. Frank, A. Klar, Radiative heat transfer and applications for glass production processes, in Mathematical Models in the Manufacturing of Glass. Lecture Notes in Mathematics (Springer, Berlin/Heidelberg, 2011), pp. 57–134Google Scholar
  9. 9.
    M. Hintermüller, K. Kunisch, Path-following methods for a class of constrained minimization problems in function space. SIAM J. Optim. 17(1), 159–187 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    M. Hintermüller, K. Kunisch, PDE-constrained optimization subject to pointwise constraints on the control, the state, and its derivative. SIAM J. Optim. 20(3), 1133–1156 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    D. Hömberg, C. Meyer, J. Rehberg, W. Ring, Optimal control for the thermistor problem. SIAM J. Control Optim. 48(5), 3449–3481 (2010)CrossRefzbMATHGoogle Scholar
  12. 12.
    K. Ito, K. Kunisch, Semi-smooth Newton methods for state-constrained optimal control problems. Syst. Control Lett. 50(3), 221–228 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    C.T. Kelley, Iterative Methods for Optimization (SIAM, Philadelphia, 1999)CrossRefzbMATHGoogle Scholar
  14. 14.
    A. Klar, E.W. Larsen, G. Thömmes, New frequency-averaged approximations to the equations of radiative heat transfer. SIAM J. Appl. Math. 64(2), 565–582 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    K. Krumbiegel, I. Neitzel, A. Rösch, Sufficient optimality conditions for the Moureau-Yosida-type regularization concept applied to semilinear elliptic optimal control problems with pointwise state constraints. Math. Appl. Ann. AOSR 2(2), 222–246 (2010)zbMATHGoogle Scholar
  16. 16.
    K. Krumbiegel, I. Neitzel, A. Rösch, Regularization for semilinear elliptic optimal control problems with pointwise state and control constraints. Comput. Optim. Appl. 52(1), 181–207 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    J. Lang, Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems (Springer, Berlin/New York, 2001)CrossRefzbMATHGoogle Scholar
  18. 18.
    J. Lang, D. Teleaga, Towards a fully space-time adaptive FEM for magnetoquasistatics. IEEE Trans. Magn. 44, 1238–1241 (2008)CrossRefGoogle Scholar
  19. 19.
    J. Lang, J. Verwer, ROS3P – an accurate third-order Rosenbrock solver designed for parabolic problems. BIT Numer. Math. 41(4), 731–738 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    J. Lang, J. Verwer, On global error estimation and control for initial value problems. SIAM J. Sci. Comput. 29, 1460–1475 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    J. Lang, J. Verwer, W-methods in optimal control. Numer. Math. 124, 337–360 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    E.W. Larsen, G. Thömmes, A. Klar, M. Seaïd, T. Götz, Simplified PN approximations to the equations of radiative heat transfer and applications. J. Comput. Phys. 183, 652–675 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    C. Meyer, I. Yousept, State-constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions. SIAM J. Control Optim. 48(2), 734–755 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    I. Neitzel, F. Tröltzsch, On convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints. Control Cybern. 37(4), 1013–1043 (2008)zbMATHGoogle Scholar
  25. 25.
    R. Pinnau, Analysis of optimal boundary control for radiative heat transfer modeled by the SPn-system. Commun. Math. Sci. 5(4), 951–969 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    R. Pinnau, A. Schulze, Newton’s method for optimal temperature-tracking of glass cooling processes. IPSE 15(4), 303–323 (2007)zbMATHMathSciNetGoogle Scholar
  27. 27.
    R. Pinnau, G. Thömmes, Optimal boundary control of glass cooling processes. Math. Methods Appl. Sci. 27(11), 1261–1281 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    L.F. Shampine, Tolerance proportionality in ODE codes, in Numerical Methods for Ordinary Differential Equations. Lecture Notes in Mathematics, vol. 1386 (Springer, Berlin/Heidelberg, 1989), pp. 118–136Google Scholar
  29. 29.
    J.C. Ziems, Adaptive multilevel SQP-methods for PDE-constrained optimization. PhD thesis, Technische Universität Darmstadt, 2010. Verlag Dr. HutGoogle Scholar
  30. 30.
    J.C. Ziems, Adaptive multilevel inexact SQP-methods for PDE-constrained optimization with control constraints. SIAM J. Optim. 23(2), 1257–1283 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    J.C. Ziems, S. Ulbrich, Adaptive multilevel generalized SQP-methods for PDE-constrained optimization. Technical report, Department of Mathematics, TU Darmstadt, 2011, submittedGoogle Scholar
  32. 32.
    J.C. Ziems, S. Ulbrich, Adaptive multilevel inexact SQP-methods for PDE-constrained optimization. SIAM J. Optim. 21(1), 1–40 (2011)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Stefanie Bott
    • 1
  • Debora Clever
    • 2
  • Jens Lang
    • 1
  • Stefan Ulbrich
    • 1
    Email author
  • Jan Carsten Ziems
    • 2
  • Dirk Schröder
    • 2
  1. 1.Department of MathematicsGraduate School of Computational Engineering Technische Universität DarmstadtDarmstadtGermany
  2. 2.Department of MathematicsTechnische Universität DarmstadtDarmstadtGermany

Personalised recommendations