Mathematical Programming

, Volume 136, Issue 1, pp 209–227 | Cite as

A note on the implementation of an interior-point algorithm for nonlinear optimization with inexact step computations

  • Frank E. Curtis
  • Johannes Huber
  • Olaf Schenk
  • Andreas Wächter
Full Length Paper Series B


This paper describes an implementation of an interior-point algorithm for large-scale nonlinear optimization. It is based on the algorithm proposed by Curtis et al. (SIAM J Sci Comput 32:3447–3475, 2010), a method that possesses global convergence guarantees to first-order stationary points with the novel feature that inexact search direction calculations are allowed in order to save computational expense. The implementation follows the proposed algorithm, but includes many practical enhancements, such as functionality to avoid the computation of a normal step during every iteration. The implementation is included in the IPOPT software package paired with an iterative linear system solver and preconditioner provided in PARDISO. Numerical results on a large nonlinear optimization test set and two PDE-constrained optimization problems with control and state constraints are presented to illustrate that the implementation is robust and efficient for large-scale applications.


Large-scale optimization PDE-constrained optimization Interior-point methods Nonconvex programming Line search Trust regions Inexact linear system solvers Krylov subspace methods 

Mathematics Subject Classification

49M05 49M15 49M37 65F10 65K05 65N22 90C06 90C26 90C30 90C51 90C90 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arnautu V., Neittaanmaki P.: Optimal Control from Theory to Computer Programs. Kluwer, Dordrecht (2003)zbMATHGoogle Scholar
  2. 2.
    Balay, S., Brown, J., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc users manual. Technical Report ANL-95/11—Revision 3.1, Argonne National Laboratory (2010)Google Scholar
  3. 3.
    Betts J.T.: Practical Methods for Optimal Control Using Nonlinear Programming Advances in Design and Control. SIAM, Philadelphia (2001)Google Scholar
  4. 4.
    Biegler L.T., Ghattas O., Heinkenschloss M., Keyes D., Bloemen Waanders B.: Real-Time PDE-Constrained Optimization. Computational Science and Engineering. SIAM, Philadelphia (2007)CrossRefGoogle Scholar
  5. 5.
    Biegler, L.T., Ghattas, O., Heinkenschloss, M., Van Bloemen Waanders, B. (eds.): Large-Scale PDE-Constrained Optimization. Lecture Notes in Computational Science and Engineering. Springer, Berlin (2003)Google Scholar
  6. 6.
    Biros G., Ghattas O.: Parallel Lagrange–Newton–Krylov–Schur methods for PDE-constrained optimization. Part I: the Krylov–Schur solver. SIAM J. Sci. Comput. 27(2), 687–713 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Biros G., Ghattas O.: Parallel Lagrange–Newton–Krylov–Schur methods for PDE-constrained optimization. Part II: the Lagrange–Newton solver and its application to optimal control of steady viscous flows. SIAM J. Sci. Comput. 27(2), 714–739 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Byrd R.H., Curtis F.E., Nocedal J.: An inexact SQP method for equality constrained optimization. SIAM J. Optim. 19(1), 351–369 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Byrd R.H., Curtis F.E., Nocedal J.: An inexact Newton method for nonconvex equality constrained optimization. Math. Program. 122(2), 273–299 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Curtis, F.E., Huber, J., Schenk, O., Wächter, A.: On the implementation of an interior-point algorithm for nonlinear optimization with inexact step computations. Technical report, Lehigh ISE 12T-006, Optimization Online ID: 2011-04-2992 (2012)Google Scholar
  11. 11.
    Curtis F.E., Nocedal J.: Flexible penalty functions for nonlinear constrained optimization. IMA J. Numer. Anal. 28(4), 749–769 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Curtis F.E., Nocedal J., Wächter A.: A matrix-free algorithm for equality constrained optimization problems with rank deficient Jacobians. SIAM J. Optim. 20(3), 1224–1249 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Curtis F.E., Schenk O., Wächter A.: An interior-point algorithm for nonlinear optimization with inexact step computations. SIAM J. Sci. Comput. 32(6), 3447–3475 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Dembo R.S., Eisenstat S.C., Steihaug T.: Inexact Newton methods. SIAM J. Numer. Anal. 19(2), 400–408 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Fourer R., Gay D.M., Kernighan B.W.: AMPL: A Modeling Language for Mathematical Programming. Brooks/Cole, Belmont (2002)Google Scholar
  16. 16.
    Freund, R.W.: Preconditioning of symmetric, but highly indefinite linear systems. In: Sydow, A. (ed.) 15th IMACS World Congress on Scientific Computation, Modelling and Applied Mathematics, pp. 551–556. Wissenschaft & Technik, Berlin (1997)Google Scholar
  17. 17.
    Gould N.I.M., Bongartz I., Conn A.R., Toint Ph.L.: CUTE: constrained and unconstrained testing environment. ACM Trans. Math. Softw. 21(1), 123–160 (1995)zbMATHCrossRefGoogle Scholar
  18. 18.
    Haber E., Ascher U.M.: Preconditioned all-at-once methods for large, sparse parameter estimation problems. Inverse Probl. 17, 1847–1864 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Heinkenschloss M., Vicente L.N.: Analysis of inexact trust-region SQP algorithms. SIAM J. Optim. 12(2), 283–302 (2002)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Hinze M., Pinnau R., Ulbrich M., Ulbrich S.: Optimization with PDE Constraints, Volume 23 of Mathematical Modeling: Theory and Applications. Springer, Dordrecht (2009)Google Scholar
  21. 21.
    Jäger H., Sachs E.W.: Global convergence of inexact reduced SQP methods. Optim. Methods Softw. 7(2), 83–110 (1997)zbMATHCrossRefGoogle Scholar
  22. 22.
    Kirk B.S., Peterson J.W., Stogner R.H., Carey G.F.: libMesh: A C++ library for parallel adaptive mesh refinement/coarsening simulations. Eng. Comput. 22(3–4), 237–254 (2006)CrossRefGoogle Scholar
  23. 23.
    Kirk D.E.: Optimal Control Theory: An Introduction. Prentice-Hall, Englewood Cliffs (1970)Google Scholar
  24. 24.
    Powell M.J.D: A hybrid method for nonlinear equations. In: Rabinowitz, P. (ed.) Numerical methods for nonlinear algebraic equations, pp. 87–114. Gordon and Breach, London (1970)Google Scholar
  25. 25.
    Schenk O., Bollhöfer M., Roemer R.A.: On large scale diagonalization techniques for the Anderson model of localization. SIAM J. Sci. Comput. 28(3), 963–983 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Steihaug T.: The conjugate gradient method and trust regions in large scale optimization. SIAM J. Numer. Anal. 20(3), 626–637 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Tröltzsch F.: Optimal control of partial differential equations: theory, methods, and applications, Volume 112. American Mathematical Society, Providence (2010)Google Scholar
  28. 28.
    Wächter A., Biegler L.T.: On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer and Mathematical Optimization Society 2012

Authors and Affiliations

  • Frank E. Curtis
    • 1
  • Johannes Huber
    • 2
  • Olaf Schenk
    • 3
  • Andreas Wächter
    • 4
  1. 1.Department of Industrial and Systems EngineeringLehigh UniversityBethlehemUSA
  2. 2.Department of Mathematics and Computer ScienceUniversity of BaselBaselSwitzerland
  3. 3.Institute of Computational ScienceUniversitá della Svizzera italianaLuganoSwitzerland
  4. 4.Department of Industrial Engineering and Management SciencesNorthwestern UniversityEvanstonUSA

Personalised recommendations