Mathematical Programming

, Volume 117, Issue 1–2, pp 435–485 | Cite as

Primal-dual interior-point methods for PDE-constrained optimization

  • Michael UlbrichEmail author
  • Stefan Ulbrich


This paper provides a detailed analysis of a primal-dual interior-point method for PDE-constrained optimization. Considered are optimal control problems with control constraints in L p . It is shown that the developed primal-dual interior-point method converges globally and locally superlinearly. Not only the easier L -setting is analyzed, but also a more involved L q -analysis, q < ∞, is presented. In L , the set of feasible controls contains interior points and the Fréchet differentiability of the perturbed optimality system can be shown. In the L q -setting, which is highly relevant for PDE-constrained optimization, these nice properties are no longer available. Nevertheless, a convergence analysis is developed using refined techniques. In parti- cular, two-norm techniques and a smoothing step are required. The L q -analysis with smoothing step yields global linear and local superlinear convergence, whereas the L -analysis without smoothing step yields only global linear convergence.


Primal-dual interior point methods PDE-constraints Optimal control Control constraints Superlinear convergence Global convergence 

Mathematics Subject Classification (2000)

90C51 90C48 49M15 65K10 


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  1. 1.
    Allgower E., Böhmer K., Potra F., Rheinboldt W. (1986). A mesh-independence principle for operator equations and their disetizations. SIAM J. Numer. Anal. 23: 160–169 CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Alt W. (1998). Disetization and mesh-independence of Newton’s method for generalized equations. In: Fiacco, A. (eds) Mathematical programming with data perturbations. Lecture notes in pure and applied mathamatics 195, pp 1–30. Dekker, New York Google Scholar
  3. 3.
    Bergounioux M., Haddou M., Hintermüller M., Kunisch K. (2000). A comparison of a moreau-yosida based active set strategy and interior point methods for constrained optimal control problems. SIAM J. Optim. 11: 495–521 CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Bergounioux M., Ito K., Kunisch K. (1999). Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37(4): 1176–1194 CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Bonnans J.F., Shapiro A. (1998). Optimization problems with perturbations: a guided tour. SIAM Rev. 40(2): 228–264 CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    Forsgren A., Gill P.E., Wright M.H. (2002). Interior methods for nonlinear optimization. SIAM Rev. 44: 525–597 CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Hintermüller M., Ito K., Kunisch K. (2003). The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13(3): 865–888 CrossRefzbMATHGoogle Scholar
  8. 8.
    Hintermüller M., Ulbrich M. (2004). A mesh-independence result for semismooth Newton methods. Math. Program. Ser. B 101(1): 151–184 CrossRefzbMATHGoogle Scholar
  9. 9.
    Hinze, M., Kunisch, K.: Second order methods for optimal control of time-dependent fluid flow. SIAM J. Control Optim. 40(3), 925–946 (electronic) (2001)Google Scholar
  10. 10.
    Jost J. (1998). Postmodern Analysis. Springer, Heidelberg zbMATHGoogle Scholar
  11. 11.
    Kelley C.T., Sachs E.W. (1994). Multilevel algorithms for constrained compact fixed point problems. SIAM J. Sci. Comput. 15: 645–667 CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Mittelmann H.D., Maurer H. (2000). Solving elliptic control problems with interior point and SQP methods: Control and state constraints. J. Comp. Appl. Math. 120: 175–195 CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Prüfert, U., Tröltzsch, F., Weiser, M.: The convergence of an interior point method for an elliptic control problem with mixed control-state constraints. Comp. Opt. Appl. (to appear)Google Scholar
  14. 14.
    Kunisch K., los Reyes J.C. (2005). A semi-smooth Newton method for control constrained boundary optimal control of the Navier-Stokes equations. Nonlinear Anal. 62(7): 1289–1316 CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Robinson S.M. (1976). Stability theory for systems of inequalities. II: Differentiable nonlinear systems. SIAM J. Numer. Anal. 13: 497–513 CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Schiela, A., Weiser, M.: Superlinear convergence of the control reduced interior point method for PDE constrained optimization. Comp. Opt. Appl. (to appear)Google Scholar
  17. 17.
    Tröltzsch F. (2005). Optimale Steuerung partieller Differentialgleichungen: Theorie, Verfahren und Anwendungen. Vieweg, Berlin zbMATHGoogle Scholar
  18. 18.
    Ulbrich M. (2002). Nonsmooth Newton-like methods for variational inequalities and constrained optimization problems in function spaces. Habilitationsschrift, Zentrum Mathematik, TU München Google Scholar
  19. 19.
    Ulbrich M. (2003). Constrained optimal control of Navier-Stokes flow by semismooth Newton methods. Syst. Control Lett. 48(3–4): 297–311 CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Ulbrich M. (2003). Semismooth Newton methods for operator equations in function spaces. SIAM J.Optim. 13(3): 805–841 CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Ulbrich M., Ulbrich S. (1999). Global convergence of trust-region interior-point algorithms for infinite-dimensional nonconvex minimization subject to pointwise bounds. SIAM J. Control Optim. 37(3): 731–764 CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Ulbrich M., Ulbrich S. (2000). Superlinear convergence of affine-scaling interior-point Newton methods for infinite-dimensional nonlinear problems with pointwise bounds. SIAM J. Control Optim. 38(6): 1938–1984 CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Weiser M. (2005). Interior point methods in function space. SIAM J. Control Optim. 44(5): 1766–1786 CrossRefMathSciNetGoogle Scholar
  24. 24.
    Weiser, M., Deuflhard, P.: Inexact central path following algorithms for optimal control problems. SIAM J. Control Optim. (to appear)Google Scholar
  25. 25.
    Weiser, M., Gänzler, T., Schiela, A.: A control reduced primal interior point method for a class of control constrained optimal control problems. Comp. Opt. Appl. (to appear)Google Scholar
  26. 26.
    Weiser M., Schiela A. (2004). Function space interior point methods for PDE constrained optimization. PAMM 4(1): 43–46 CrossRefGoogle Scholar
  27. 27.
    Zeidler E. (1985). Nonlinear functional analysis and its applications III. Springer, New York zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Chair of Mathematical OptimizationZentrum Mathematik M1, TU MünchenGarching b. MünchenGermany
  2. 2.TU Darmstadt, Fachbereich MathematikAG10: Nonlinear Optimization and Optimal ControlDarmstadtGermany

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