Preconditioners for Karush-Kuhn-Tucker Matrices Arising in the Optimal Control of Distributed Systems

  • A. Battermann
  • M. Heinkenschloss
Part of the International Series of Numerical Mathematics book series (ISNM, volume 126)


In this paper preconditioners for linear systems arising in interior-point methods for the solution of distributed control problems are derived and analyzed. The matrices K in these systems have a block structure with blocks obtained from the discretization of the objective function and the governing differential equation. The preconditioners have a block structure with blocks being composed of preconditioners for the subblocks of the system matrix K. The effectiveness of the preconditioners is analyzed and numerical examples for an elliptic model problem are shown.

1991 Mathematics Subject Classification

49M30 49N10 90C06 90C20 

Key words and phrases

Preconditioners iterative methods interior point methods linearquadratic optimal control problems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter Systems, Systems & Control: Foundations & Applications, Birkhôuser-Verlag, Boston, Basel, Berlin, 1989.Google Scholar
  2. 2.
    R. Barrett, M. Berry, T. F. Chan, J. D. J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, AND H. van der Vorst, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1993.MATHGoogle Scholar
  3. 3.
    A. Battermann,Preconditioners for Karush-Kuhn-Tucker systems arising in optimal control, Master’s thesis, Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 1996.Google Scholar
  4. 4.
    J. Bonnans, C. Pola, and R. RebaI, Perturbed path following interior point algorithm, Tech. Rep. No. 2745, INRIA, Domaine de Voluceau, 78153 Rocquencourt, France, 1995.Google Scholar
  5. 5.
    D. Braess and P. Peisker, On the numerical solution of the biharmonic equation and the role of squaring matrices, Ima J. Numer. Anal, 6 (1986), pp. 393–404.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    L. Collatz AND W. Wetterling, Optimization Problems, Springer-Verlag, Berlin, Heidelberg, New York, 1975.CrossRefGoogle Scholar
  7. 7.
    J. E. Dennis, M. Heinkenschloss, and L. N. Vicente, Trust-region interior-point algorithms for a class of nonlinear programming problems, Tech. Rep. TR94-45, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005-1892, 1994. Available electronically at Scholar
  8. 8.
    A. S. El-Bakry, R. A. Tapia, T. Tsuchiya, and Y. Zhang, On the formulation and theory of the primal-dual Newton interior-point method for nonlinear programming, Journal of Optimization Theory and Applications, 89 (1996), pp. 507–541.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    R. W. Freund and F. Jarre, A qmr-based interior-point method for solving linear programs, Mathematical Programming, Series B, (to appear).Google Scholar
  10. 10.
    P. E. Gill, W. Murray, D. B. Ponceleön, and M. A. Saunders, Preconditioned for indefinite systems arising in optimization, Siam J. Matrix Anal. Appl., 13 (1992), pp. 292–311.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    G. Golub AND C. F. van Loan, Matrix Computations, John Hopkins University Press, Baltimore, London, 1989.MATHGoogle Scholar
  12. 12.
    S. Ito, C. T. Kelley, and E. W. Sachs, Inexact primal-dual interior-point iteration for linear programs in function spaces, Computational Optimization and Applications, 4 (1995), pp. 189–201.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Berlin, Heidelberg, New York, 1971.MATHCrossRefGoogle Scholar
  14. 14.
    C. C. Paige and M. A. Saunders, Solution of sparse indefinite systems of linear equations, Siam J. Numer. Anal, 12 (1975), pp. 617–629.MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    T. Rusten And R. Winther, A preconditioned iterative method for saddlepoint problems, SIAM J. Matrix Anal. Appl, 13 (1992), pp. 887–904.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    J.Stoer,Solution of large linear systems of equations by conjugate gradient type methods, in Mathematical Programming, The State of The Art, A.Bachem, M. Grötschel, and B.Korte, eds. Springer-Verlag, Berlin, Heidelberg, New-York, 1983, pp. 540–565.CrossRefGoogle Scholar
  17. 17.
    D. Sylvester and A. Wathen, Fast iterative solution of stabilized Stokes systems part II: using general block preconditioners, SIAM J. Numer. Anal, 31 (1994), pp. 1352–1367.CrossRefMathSciNetGoogle Scholar
  18. 18.
    L. N. Vicente, On interior point Newton algorithms for discretized optimal control problems with state constraints, tech. rep, Departamento de Matematica, Universidade de Coimbra, 3000 Coimbra, Portugal, 1996.Google Scholar
  19. 19.
    M. H. Wright, Interior point methods for constrained optimization, in Acta Numerica 1992, A. Iserles, ed, Cambridge University Press, Cambridge, London, New York, 1992, pp. 341–407.Google Scholar
  20. 20.
    S. J. Wright, Primal-Dual Interior-Point Methods, SIAM, Philadelphia, PA, 1996.Google Scholar

Copyright information

© Springer Basel AG 1998

Authors and Affiliations

  • A. Battermann
    • 1
  • M. Heinkenschloss
    • 2
  1. 1.FB IV-MathematikaUniversität TrierTrierFederal Republic of Germany
  2. 2.Department of Computational and Applied MathematicsRice UniversityHoustonUSA

Personalised recommendations