Skip to main content

Sequential quadratic programming methods for parametric nonlinear optimization

Abstract

Sequential quadratic programming (SQP) methods are known to be efficient for solving a series of related nonlinear optimization problems because of desirable hot and warm start properties—a solution for one problem is a good estimate of the solution of the next. However, standard SQP solvers contain elements to enforce global convergence that can interfere with the potential to take advantage of these theoretical local properties in full. We present two new predictor–corrector procedures for solving a nonlinear program given a sufficiently accurate estimate of the solution of a similar problem. The procedures attempt to trace a homotopy path between solutions of the two problems, staying within the local domain of convergence for the series of problems generated. We provide theoretical convergence and tracking results, as well as some numerical results demonstrating the robustness and performance of the methods.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

References

  1. 1.

    Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods. Colorado State University Press, Fort Collins (1990)

    Book  Google Scholar 

  2. 2.

    Bonnans, J.: Local analysis of Newton-type methods for variational inequalities and nonlinear programming. Appl. Math. Optim. 29, 161–186 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Bonnans, J.F., Shapiro, A.: Optimization problems with perturbations: a guided tour. SIAM Rev. 40, 228–264 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, Berlin (2000)

    Book  MATH  Google Scholar 

  5. 5.

    Diehl, M.: Real-time optimization for large scale nonlinear processes. PhD thesis, Universität Heidelberg (2001)

  6. 6.

    Facchinei, F., Fischer, A., Herrich, M.: A family of Newton methods for nonsmooth constrained systems with nonisolated solutions. Math. Methods Oper. Res. 1–11 (2011)

  7. 7.

    Facchinei, F., Fischer, A., Kanzow, C.: On the accurate identification of active constraints. SIAM J. Optim. 9, 14–32 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. 8.

    Fernández, D., Solodov, M.: Stabilized sequential quadratic programming for optimization and a stabilized Newton-type method for variational problems. Math. Program. 125, 47–73 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    Ferreau, H.: An online active set strategy for fast solution of parametric quadratic programs with applications to predictive engine control. Master’s thesis, University of Heidelberg (2006)

  10. 10.

    Ferreau, H.: qpOASES - An open-source implementation of the online active set strategy for fast model predictive control. In: Proceedings of the Workshop on Nonlinear Model Based Control: Software and Applications, Loughborough, pp. 29–30 (2007)

  11. 11.

    Ferreau, H., Kirches, C., Potschka, A., Bock, H., Diehl, M.: qpOASES: A parametric active-set algorithm for quadratic programming. Math. Program. Comput. 1–37 (2013)

  12. 12.

    Gal, T.: A historical sketch on sensitivity analysis and parametric programming. In: Gal, T., Greenberg, H. (eds.) Advances in Sensitivity Analysis and Parametric Programming. International series in operations research and management science, vol. 6, pp. 1–10. Springer, New York (1997)

    Chapter  Google Scholar 

  13. 13.

    Gfrerer, H., Guddat, J., Wacker, H.: A globally convergent algorithm based on imbedding and parametric optimization. Computing 30(3), 225–252 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    Gill, P., Kungurtsev, V., Robinson, D.: A regularized SQP method convergent to second order optimal points. Technical Report 13–04, UCSD CCoM (2013)

  15. 15.

    Gould, N., Orban, D., Toint, P.L.: CUTEst: a constrained and unconstrained testing environment with safe threads. Cahier du GERAD G 2013, 27 (2013)

    Google Scholar 

  16. 16.

    Guddat, J., Vasquez, F.G., Jongen, H.: Parametric Optimization: Singularities. Pathfollowing and Jumps. Teubner, Stuttgart (1990)

    MATH  Google Scholar 

  17. 17.

    Guddat, J., Wacker, H., Zulehner, W.: On imbedding and parametric optimizationa concept of a globally convergent algorithm for nonlinear optimization problems. In: Fiacco, A. (ed.) Sensitivity. Stability and parametric analysis, volume 21 of mathematical programming studies, pp. 79–96. Springer, Berlin (1984)

    Google Scholar 

  18. 18.

    Hager, W.W., Gowda, M.S.: Stability in the presence of degeneracy and error estimation. Math. Program. 85(1), 181–192 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

    Hock, W., Schittkowski, K.: Lecture Notes in Economics and Mathematical Systems. Test examples for nonlinear programming codes. Springer, Berlin (1981)

    Google Scholar 

  20. 20.

    Izmailov, A.: Solution sensitivity for Karush–Kuhn–Tucker systems with non-unique Lagrange multipliers. Optimization 59(5), 747–775 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. 21.

    Izmailov, A.F., Solodov, M.V.: Stabilized SQP revisited. Math. Program. 133, 93–120 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. 22.

    Johnson, T.C., Kirches, C., Wächter. A.: An active-set quadratic programming method based on sequential hot-starts. 2013. Available at optimization online.

  23. 23.

    Jongen, H.T., Jonker, P., Twilt, F.: Critical sets in parametric optimization. Math. Program. 34(3), 333–353 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  24. 24.

    Jongen, H.T., Weber, G.W.: On parametric nonlinear programming. Ann. Oper. Res. 27, 253–283 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  25. 25.

    Klatte, D., Kummer, B.: Stability properties of infima and optimal solutions of parametric optimization problems. In: Demyanov, V., Pallaschke, D. (eds.) Nondifferentiable Optimization: Motivations and Applications. Lecture notes in economics and mathematical systems, vol. 255, pp. 215–229. Springer, Berlin (1985)

    Chapter  Google Scholar 

  26. 26.

    Kojima, M.: Strongly stable stationary solutions in nonlinear programs. In: Robinson, S. M. (ed.) Analysis and Computation of Fixed Points, pp. 93–138. Academic Press, New York (1980)

  27. 27.

    Kungurtsev, V.: Second Derivative SQP Methods. PhD thesis, UC-San Diego (2013)

  28. 28.

    Kyparisis, J.: On uniqueness of Kuhn–Tucker multipliers in nonlinear programming. Math. Program. 32(2), 242–246 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  29. 29.

    Levy, A.B.: Solution sensitivity from general principles. SIAM J. Control Optim. 40, 1–38 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  30. 30.

    Levy, A.B., Rockafellar, R.: Advances in Nonsmooth Optimization. Sensitivity of solutions in nonlinear programs with nonunique multipliers. World Scientific Publishing, Singapore (1995)

    Google Scholar 

  31. 31.

    Lundberg, B.N., Poore, A.B.: Numerical continuation and singularity detection methods for parametric nonlinear programming. SIAM J. Optim. 3, 134–154 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  32. 32.

    Mostafa, E.-S.M., Vicente, L.N., Wright, S.J.: Global Optimization and Constraint Satisfaction. Numerical behavior of a stabilized SQP method for degenerate NLP problems, pp. 123–141. Springer, Berlin (2003)

    Book  Google Scholar 

  33. 33.

    Poore, A., Tiahrt, C.: Bifurcation problems in nonlinear parametric programming. Math. Program. 39, 189–205 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  34. 34.

    Ralph, D., Dempe, S.: Directional derivatives of the solution of a parametric nonlinear program. Math. Program. 70, 159–172 (1995)

    MATH  MathSciNet  Google Scholar 

  35. 35.

    Robinson, S.: Perturbed Kuhn–Tucker points and rates of convergence for a class of nonlinear programming algorithms. Math. Program. 7, 1–16 (1974)

    Article  MATH  Google Scholar 

  36. 36.

    Robinson, S.M.: Stability theory for systems of inequalities. Part I: linear systems. SIAM J. Numer. Anal. 12, 754–769 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  37. 37.

    Robinson, S.M.: Stability theory for systems of inequalities, Part II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13(4), 497–513 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  38. 38.

    Robinson, S.M.: Optimality and Stability in Mathematical Programming. Generalized equations and their solutions, part II: applications to nonlinear programming. Springer, Berlin (1982)

    Google Scholar 

  39. 39.

    Sequeira, S., Graellis, M., Puigjaner, L.: Real-time evolution for on-line optimization of continuous processes. Ind. Eng. Chem. Res. 41, 1815–1825 (2002)

    Article  Google Scholar 

  40. 40.

    Seydel, R.: Practical Bifurcation and Stability Analysis. Springer, New York (2010)

    Book  MATH  Google Scholar 

  41. 41.

    Tran-Dinh, Q., Savorgnan, C., Diehl, M.: Adjoint-based predictor–corrector sequential convex programming for parametric nonlinear optimization. SIAM J. Optim. 22(4), 12581284 (2012)

    Google Scholar 

  42. 42.

    Wachsmuth, G.: On LICQ and the uniqueness of Lagrange multipliers. Oper. Res. Lett. 41(1), 78–80 (2013)

  43. 43.

    Watson, L.T.: Solving the nonlinear complementarity problem by a homotopy method. SIAM J. Control Optim. 17, 36–46 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  44. 44.

    Wright, S.J.: Superlinear convergence of a stabilized SQP method to a degenerate solution. Comput. Optim. Appl. 11, 253–275 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  45. 45.

    Wright, S.J.: An algorithm for degenerate nonlinear programming with rapid local convergence. SIAM J. Optim. 15(3), 673–696 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  46. 46.

    Zavala, V., Anitescu, M.: Real-time nonlinear optimization as a generalized equation. SIAM J. Control Optim. 48(8), 5444–5467 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

We would like to thank Greg Horn for insightful discussions in regards to some of the details of the numerical software implementation of the algorithms, in particular with regards to parallelization. In addition, we’d like to thank the reviewers for their helpful suggestions for improving the presentation of the material in the paper. This research was supported by Research Council KUL: PFV/10/002 Optimization in Engineering Center OPTEC, GOA/10/09 MaNet and GOA/10/11 Global real-time optimal control of autonomous robots and mechatronic systems. Flemish Government: IOF/KP/SCORES4CHEM, FWO: PhD/postdoc Grants and projects: G.0320.08 (convex MPC), G.0377.09 (Mechatronics MPC); IWT: PhD Grants, projects: SBO LeCoPro; Belgian Federal Science Policy Office: IUAP P7 (DYSCO, Dynamical systems, control and optimization, 2012-2017); EU: FP7-EMBOCON (ICT- 248940), FP7-SADCO ( MC ITN-264735), FP7-TEMPO, ERC ST HIGHWIND (259 166), Eurostars SMART, ACCM.

It was also supported by the European social fund within the framework of realizing the project “Support of inter-sectoral mobility and quality enhancement of research teams at the Czech Technical University in Prague”, CZ.1.07/2.3.00/30.0034.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Vyacheslav Kungurtsev.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kungurtsev, V., Diehl, M. Sequential quadratic programming methods for parametric nonlinear optimization. Comput Optim Appl 59, 475–509 (2014). https://doi.org/10.1007/s10589-014-9696-2

Download citation

Keywords

  • Parametric nonlinear programming
  • Nonlinear programming
  • Nonlinear constraints
  • Sequential quadratic programming
  • SQP methods
  • Stabilized SQP
  • Regularized methods
  • Model predictive control

Mathematics Subject Classification

  • 49J20
  • 49J15
  • 49M37
  • 49D37
  • 65F05
  • 65K05
  • 90C30