Skip to main content
Log in

A trust region SQP algorithm for mixed-integer nonlinear programming

Optimization Letters Aims and scope Submit manuscript

Abstract

We propose a modified sequential quadratic programming method for solving mixed-integer nonlinear programming problems. Under the assumption that integer variables have a smooth influence on the model functions, i.e., that function values do not change drastically when in- or decrementing an integer value, successive quadratic approximations are applied. The algorithm is stabilized by a trust region method with Yuan’s second order corrections. It is not assumed that the mixed-integer program is relaxable or, in other words, function values are evaluated only at integer points. The Hessian of the Lagrangian function is approximated by a quasi-Newton update formula subject to the continuous and integer variables. Numerical results are presented for a set of 80 mixed-integer test problems taken from the literature. The surprising result is that the number of function evaluations, the most important performance criterion in practice, is less than the number of function calls needed for solving the corresponding relaxed problem without integer variables.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Audet C., Dennis J.E. (2001): Pattern search algorithm for mixed variable programming. SIAM J. Optim. 11, 573–594

    Article  MathSciNet  Google Scholar 

  2. Borchers B., Mitchell J.E. (1994): An improved branch and bound algorithm for mixed integer nonlnear programming. Comput. Oper. Res. 21(4): 359–367

    Article  MATH  MathSciNet  Google Scholar 

  3. Bünner M.J., Schittkowski K., van de Braak G. (2004): Optimal design of electronic components by mixed-integer nonlinear programming. Optim. Eng. 5, 271–294

    Article  MathSciNet  Google Scholar 

  4. Burke J.V. (1992): A robust trust region method for constrained nonlinear programming problems. SIAM J. Optim. 2, 325–347

    Article  MATH  MathSciNet  Google Scholar 

  5. Byrd R., Schnabel R.B., Schultz G.A. (1987): A trust region algorithm for nonlinearly constrained optimization. SIAM J. Numer. Anal. 24, 1152–1170

    Article  MATH  MathSciNet  Google Scholar 

  6. Celis, M.R.: A trust region strategy for nonlinear equality constrained optimization. Ph.D. Thesis, Department of Mathematics. Rice University (1983)

  7. Conn A.R., Gould I.M., Toint P.L. (2000): Trust-Region Methods. SIAM, Philadelphia

    MATH  Google Scholar 

  8. Duran M., Grossmann I.E. (1986): An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Mathe. Program. 36, 307–339

    Article  MATH  MathSciNet  Google Scholar 

  9. Exler, O., Schittkowksi, K.: MISQP: a Fortran implementation of a trust region SQP algorithm for mixed-integer nonlinear programming—user’s guide, version 2.1, Report, Department of Computer Science, University of Bayreuth (2006). http://www.uni-bayreuth.de/departments/math/~kschittkowski/MISQP.pdf

  10. Fletcher R. (1981): Practical Methods of Optimization, vol. 2, Constrained Optimization. Wiley, Chichester

    MATH  Google Scholar 

  11. Fletcher R. (1982): Second order correction for nondifferentiable optimization. In: Watson G.A. (eds) Numerical Analysis. Springer, Berlin Heidelberg New York, pp. 85–114

    Chapter  Google Scholar 

  12. Fletcher R., Leyffer S. (1994): Solving mixed integer nonlinear programs by outer approximation. Math. Program. 66, 327–349

    Article  MathSciNet  Google Scholar 

  13. Floudas C.A. (1995): Nonlinear and Mixed-Integer Optimization. Oxford University Press, New York, Oxford

    MATH  Google Scholar 

  14. Floudas, C.A., Pardalos, P.M., Adjiman, C.S., Esposito, W.R., Gumus, Z.H., Harding S.T., Klepeis, J.L., Meyer, C.A., Schweiger, C.A.: Handbook of Test Problems in Local and Global Optimization. Kluwer (1999)

  15. Gill P.E., Murray W., Wright M. (1981): Practical Optimization. Academic, New York

    MATH  Google Scholar 

  16. Grossmann I.E., Kravanja Z. (1997): Mixed-integer nonlinear programming: A survey of algorithms and applications. In: Conn A.R., Biegler L.T., Coleman T.F., Santosa F.N. (eds) Large-Scale Optimization with Applications, Part II: Optimal Design and Control. Springer, Berlin Heidelberg New York

    Google Scholar 

  17. Gupta O.K., Ravindran V. (1985): Branch and bound experiments in convex nonlinear integer programming. Manag. Sci. 31, 1533–1546

    Article  MATH  MathSciNet  Google Scholar 

  18. Leyffer S. (2001): Integrating SQP and branch-and-bound for mixed integer nonlinear programming. Comput. Optim. Appl. 18, 295–309

    Article  MATH  MathSciNet  Google Scholar 

  19. Li H.-L., Chou C.-T. (1994): A global approach for nonlinear mixed discrete programming in design optimization. Eng. Optim. 22, 109–122

    Google Scholar 

  20. Moré J.J. (1983): Recent developments in algorithms and software for trust region methods. In: Bachem A., Grötschel M., Korte B. (eds) Mathematical Programming: The State of the Art. Springer, Berlin Hedelberg New York, pp. 258–287

    Google Scholar 

  21. Powell M.J.D. (1984): On the global convergence of trust region algorithms for unconstrained minimization. Math. Program. 29, 297–303

    Article  MATH  MathSciNet  Google Scholar 

  22. Powell M.J.D., Yuan Y. (1991): A trust region algorithm for equality constrained optimization. Math. Program. 49, 189–211

    Article  MathSciNet  Google Scholar 

  23. Schittkowski, K.: QL: A Fortran code for convex quadratic programming—user’s guide, version 2.11. Report, Department of Mathematics, University of Bayreuth (2003) http://www.uni-bayreuth.de/departments/math/~kschittkowski/QL.pdf

  24. Spickenreuther, T.: Entwicklung eines allgemeinen Branch & Bound Ansatzes zur gemischt-ganzzahligen Optimierung. Diploma Thesis, Department of Mathematics, University of Bayreuth (2005)

  25. Stoer, J.: Foundations of recursive quadratic programming methods for solving nonlinear programs. In: Schittkowski, K. (ed.) Computational Mathematical Programming, vol. 15. NATO ASI Series, Series F: Computer and Systems Sciences. Springer, Berlin Heidelberg New York (1985)

  26. Toint P.L. (1997): A nonmonotone trust-region algorithm for nonlinear optimization subject to convex constraints. Math. Program. 77, 69–94

    MathSciNet  Google Scholar 

  27. Westerlund P. (2002): Solving pseudo-convex mixed integer optimization problems by cutting plane techniques. Optim. Eng. 3, 253–280

    Article  MATH  MathSciNet  Google Scholar 

  28. Yuan Y.-X. (1984): An example of only linearly convergence of trust region algorithms for nonsmooth optimization. IMA J. Numer. Anal. 4, 327–335

    Article  MATH  MathSciNet  Google Scholar 

  29. Yuan Y.-X. (1985): On the superlinear convergence of a trust region algorithm for nonsmooth optimization. Math. Program. 31, 269–285

    Article  MATH  Google Scholar 

  30. Yuan Y.-X. (1995): On the convergence of a new trust region algorithm. Numer. Math. 70, 515–539

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Klaus Schittkowski.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Exler, O., Schittkowski, K. A trust region SQP algorithm for mixed-integer nonlinear programming. Optimization Letters 1, 269–280 (2007). https://doi.org/10.1007/s11590-006-0026-1

Download citation

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-006-0026-1

Keywords

Navigation