Mathematical Programming Computation

, Volume 4, Issue 4, pp 307–331 | Cite as

Optimal sensitivity based on IPOPT

  • Hans Pirnay
  • Rodrigo López-Negrete
  • Lorenz T. Biegler
Full Length Paper


We introduce a flexible, open source implementation that provides the optimal sensitivity of solutions of nonlinear programming (NLP) problems, and is adapted to a fast solver based on a barrier NLP method. The program, called sIPOPT evaluates the sensitivity of the Karush–Kuhn–Tucker (KKT) system with respect to perturbation parameters. It is paired with the open-source IPOPT NLP solver and reuses matrix factorizations from the solver, so that sensitivities to parameters are determined with minimal computational cost. Aside from estimating sensitivities for parametric NLPs, the program provides approximate NLP solutions for nonlinear model predictive control and state estimation. These are enabled by pre-factored KKT matrices and a fix-relax strategy based on Schur complements. In addition, reduced Hessians are obtained at minimal cost and these are particularly effective to approximate covariance matrices in parameter and state estimation problems. The sIPOPT program is demonstrated on four case studies to illustrate all of these features.


NLP Sensitivity Interior point 

Mathematics Subject Classification (2000)

90C30 90C31 90C51 90-08 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Wächter A., Biegler L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106(1), 25–57 (2006)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Fiacco A.V.: Introduction to sensitivity and stability analysis in nonlinear programming. Mathematics in Science and Engineering, vol. 165. Academic Press, Dublin (1983)Google Scholar
  3. 3.
    Fiacco A.V., Ishizuka Y.: Sensitivity and stability analysis for nonlinear programming. Ann. Oper. Res. 27, 215–236 (1990)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Büskens H., Maurer C.: Sensitivity analysis and real-time control of parametric control problems using nonlinear programming methods. In: Grötschel, M., Krumke, S., Rambau, J. (eds) Online Optimization of Large-scale Systems, pp. 57–68. Springer, Berlin (2001)Google Scholar
  5. 5.
    Kyparisis J.: Sensitivity analysis for nonlinear programs and variational inequalities with nonunique multipliers. Math. Oper. Res. 15(2), 286–298 (1990)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Kojima M.: Strongly stable stationary solutions in nonlinear programs. In: Robinson, S.M. (eds) Analysis and Computation of Fixed Points, pp. 93–138. Academic Press, New York (1980)Google Scholar
  7. 7.
    Kojima M., Hirabayashi R.: Continuous deformation of nonlinear programs. Math. Program. Study 21, 150–198 (1984)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Jongen H.T., Jonker P., Twilt F.: Nonlinear Optimization in Finite Dimensions. Kluwer, Dordrecht (2000)MATHGoogle Scholar
  9. 9.
    Jongen H.T., Meer K., Triesch E.: Optimization Theory. Kluwer, Dordrecht (2004)MATHGoogle Scholar
  10. 10.
    Fiacco, A.V. Ghaemi, A.: A user’s manual for SENSUMT. A penalty function computer program for solution, sensitivity analysis and optimal bound value calculation in parametric nonlinear programs. Technical Report T-434, Management Science and Engineering, George Washington University (1980)Google Scholar
  11. 11.
    Ganesh N., Biegler L.T.: A reduced hessian strategy for sensitivity analysis of optimal flowsheets. AIChE 33, 282–296 (1987)CrossRefGoogle Scholar
  12. 12.
    Wolbert D., Joulia X., Koehret B., Biegler L.T.: Flowsheet optimization and optimal sensitivity analysis using exact derivatives. Comput. Chem. Eng. 18, 1083 (1994)CrossRefGoogle Scholar
  13. 13.
    Forbes J., Marlin T.E.: Design cost: a systematic approach to technology selection for model-based real-time optimization systems. Comput. Chem. Eng. 20, 717–734 (1996)CrossRefGoogle Scholar
  14. 14.
    Diehl M., Findeisen R., Allgöwer F.: A stabilizing real-time implementation of nonlinear model predictive control. In: Biegler, L.T., Keyes, D., Ghattas, O., van Bloemen Waanders, B., Heinkenschloss, M. (eds) Real-Time PDE-Constrained Optimization, pp. 25–52. SIAM, Philadelphia (2007)CrossRefGoogle Scholar
  15. 15.
    Kadam, J. Marquardt, W.: Sensitivity-based solution updates in closed-loop dynamic optimization. In: Proceedings of the DYCOPS 7 Conference. Elsevier, Amsterdam (2004)Google Scholar
  16. 16.
    Zavala V.M., Biegler L.T.: The advanced-step NMPC controller: optimality, stability and robustness. Automatica 45(1), 86–93 (2009)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Forsgren A., Gill P.E., Wright M.H.: Interior point methods for nonlinear optimization. SIAM Rev. 44(4), 525–597 (2002)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Guddat J., Guerra Vazquez F., Jongen H.T.: Parametric Optimization: Singularities, Pathfollowing and Jumps. Teubner, Stuttgart (1990)MATHGoogle Scholar
  19. 19.
    Robinson S.M.: Generalized equations and their solutions, part II: applications to nonlinear programming. Math. Program. Study 19, 200–221 (1982)MATHCrossRefGoogle Scholar
  20. 20.
    Diehl, M.: Real-Time Optimization for Large Scale Nonlinear Processes. Ph.D. thesis, Universität Heidelberg (2001).
  21. 21.
    Zavala, V.M.: Computational Strategies for the Operation of Large-Scale Chemical Processes. Ph.D. thesis, Carnegie Mellon University (2008)Google Scholar
  22. 22.
    Bartlett R.A., Biegler L.T.: QPSchur: a dual, active-set, schur-complement method for large-scale and structured convex quadratic programming. Optim. Eng. 7, 5–32 (2006)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Fourer R., Gay D.M., Kernighan B.W.: AMPL: a modeling language for mathematical programming. Duxbury Press, Pacific Grove (2002)Google Scholar
  24. 24.
    Pirnay, H., López-Negrete, R., Biegler, L.T.: sIPOPT Reference Manual. Carnegie Mellon University (2011).
  25. 25.
    Rajaraman S., Hahn J., Mannan M.S.: A methodology for fault detection, isolation, and identification for nonlinear processes with parametric uncertainties. Ind. Eng. Chem. Res. 43(21), 6774–6786 (2004)CrossRefGoogle Scholar
  26. 26.
    Biegler L.T.: Nonlinear Programming: Concepts, Algorithms, and Applications to Chemical Processes. SIAM, Philadelphia (2010)MATHCrossRefGoogle Scholar
  27. 27.
    López-Negrete R., Flores-Tlacuahuac A.: Optimal start-up and product transition policies of a reactive distillation column. Ind. Eng. Chem. Res. 46, 2092–2111 (2007)CrossRefGoogle Scholar
  28. 28.
    Cervantes A.M., Biegler L.T.: Large-scale DAE optimization using a simultaneous NLP formulation. AIChE J. 44(5), 1038–1050 (1998)CrossRefGoogle Scholar
  29. 29.
    Bard Y.: Nonlinear Parameter Estimation. Academic Press, New York (1974)MATHGoogle Scholar

Copyright information

© Springer and Mathematical Optimization Society 2012

Authors and Affiliations

  • Hans Pirnay
    • 1
  • Rodrigo López-Negrete
    • 2
  • Lorenz T. Biegler
    • 2
  1. 1.AVT-Process Systems EngineeringRWTH Aachen UniversityAachenGermany
  2. 2.Department of Chemical EngineeringCarnegie Mellon UniversityPittsburghUSA

Personalised recommendations