Skip to main content
SpringerLink
Log in

A comparative study of SQP-type algorithms for nonlinear and nonconvex mixed-integer optimization

  • Full Length Paper
  • Published:
Mathematical Programming Computation Aims and scope Submit manuscript

Abstract

We present numerical results of a comparative study of codes for nonlinear and nonconvex mixed-integer optimization. The underlying algorithms are based on sequential quadratic programming (SQP) with stabilization by trust-regions, linear outer approximations, and branch-and-bound techniques. The mixed-integer quadratic programming subproblems are solved by a branch-and-cut algorithm. Second order information is updated by a quasi-Newton update formula (BFGS) applied to the Lagrange function for continuous, but also for integer variables. We do not require that the model functions can be evaluated at fractional values of the integer variables. Thus, partial derivatives with respect to integer variables are replaced by descent directions obtained from function values at neighboring grid points, and the number of simulations or function evaluations, respectively, is our main performance criterion to measure the efficiency of a code. Numerical results are presented for a set of 100 academic mixed-integer test problems. Since not all of our test examples are convex, we reach the best-known solutions in about 90 % of the test runs, but at least feasible solutions in the other cases. The average number of function evaluations of the new mixed-integer SQP code is between 240 and 500 including those needed for one- or two-sided approximations of partial derivatives or descent directions, respectively. In addition, we present numerical results for a set of 55 test problems with some practical background in petroleum engineering.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Asaadi J.: A computational comparison of some non-linear programs. Math. Program. 4, 144–154 (1973)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  3. Ayatollahi S., Narimani M., Moshfeghiam M.: Intermittent gas lift in Aghjari Oil 488 Field, a mathematical study. J. Petroleum Sci. Eng. 42, 245–255 (2004)

    Article  Google Scholar 

  4. Bellman R.: Introduction to Matrix Analysis. McGraw-Hill, India (1960)

    MATH  Google Scholar 

  5. Belotti P., Lee J., Liberti L., Margot F., Wächter A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24, 597–634 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Belotti, P.: Couenne: a user’s manual, Technical Report, Department of Mathematical Sciences, Clemson University, Clemson SC 29643, USA (2009)

  7. Bonami, P., Biegler, L.T., Conn, A.R., Cornuejols, G., Grossmann, I.E., Laird, C.D., Lee, J., Lodi, A., Margot, F., Sawaya, N., Waechter, A.: An algorithmic framework for convex mixed integer nonlinear programs. IBM Research Report RC23771 (2005)

  8. Bonami, P., Kilinç, M., Linderoth, J.: Algorithms and software for convex mixed-integer nonlinear programs, Technical Report No. 1664, Computer Science Department, University of Wisconsin, Madison (2009)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Bussieck, M.R., Vigerske, S. : MINLP Solver Software, submitted for publication (2010)

  12. Bussieck, M.R., Drud, A.S., Meeraus, A.: MINLPLib—a collection of test models for mixed integer nonlinear programming. GAMS Development Corp, Washington D.C., USA (2007)

  13. Chen Z., Zhang X.: A nonmonotone trust-region algorithm with nonmonotone penalty parameters for constrained optimization. J. Comput. Appl. Math. 172, 7–39 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Conn, A.R., Gould, N.M., Toint, P.L.: Trust-region methods. MPS-SIAM Series on Optimization, Philadelphia (2000)

  15. Deng N.Y., Xiao Y., Zhou F.J.: Nonmonotonic trust-region algorithm. J. Optim. Theory Appl. 26, 259–285 (1993)

    Article  MathSciNet  Google Scholar 

  16. Dolan E.D., Moré J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–213 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Duran M., Grossmann I.E.: An outer-approximation algorithm for a class of Mixed Integer Nonlinear Programs. Math. Program. 36, 307–339 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  18. Exler O., Antelo L.T., Egea J.A., Alonso A.A., Banga J.R.: Tabu search-based algorithm for mixed-integer nonlinear problems and its application to integrated process and control system design. Comput. Chem. Eng. 32(8), 1877–1891 (2008)

    Article  Google Scholar 

  19. Exler, O., Lehmann, T., Schittkowski, K.: MISQPN : a Fortran subroutine for mixed-integer nonlinear optimization by outer approximation supported by mixed-integer search steps - user’s guide, version 1.0, Report, Department of Computer Science, University of Bayreuth (2009)

  20. Exler, O., Lehmann, T., Schittkowski, K.: MISQP: A Fortran subroutine of a trust region SQP algorithm for mixed-integer nonlinear programming—user’s guide, Report, Department of Computer Science, University of Bayreuth (2012)

  21. Exler O., Schittkowski K.: A trust region SQP algorithm for mixed integer nonlinear programming. Optim. Lett. 1(3), 269–280 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Fletcher R.: A new approach to variable metric algorithms. Comput. J. 13, 317–322 (1970)

    Article  MATH  Google Scholar 

  23. Fletcher, R.: Second order correction for nondifferentiable optimization. In: Watson, G.A. (ed.) Numerical analysis, pp. 85–114, Springer, Berlin (1982)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  26. Fukushima M.: A successive quadratic programming algorithm with global and superlinear convergence properties. Math. Program. 35, 253–264 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  27. Goldfarb D., Idnani A.: A numerically stable method for solving strictly convex quadratic programs. Math. Program. 27, 1–33 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  28. Grossmann I.E.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3, 227–252 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. Grossmann, I.E., Kravanja, Z.: 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, New York (1997)

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Günlük O., Linderoth J.: Perspective relaxation of mixed integer nonlinear programs with indicator variables. Math. Program. Series B 104, 186–203 (2010)

    Google Scholar 

  32. Hartwanger C., Schittkowski K., Wolf H.: Computer aided optimal design of horn radiators for satellite communication. Eng. Optim. 33, 221–244 (2000)

    Article  Google Scholar 

  33. Hock W., Schittkowski K.: A comparative performance evaluation of 27 nonlinear programming codes. Computing 30, 335–358 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lehmann, T., Schittkowski, K.: MIQL: A Fortran subroutine for convex mixed-integer quadratic programming—user’s guide, version 1.0, Report, Department of Computer Science, University of Bayreuth (2009)

  35. Lehmann, T., Schittkowski, K.: MINLPB4: A Fortran code for nonlinear mixed-integer quadratic programming by branch-and-bound - user’s guide, version 1.0, Report, Department of Computer Science, University of Bayreuth (2009)

  36. Lehmann, T., Schittkowski, K.: MISQPOA: a Fortran subroutine for mixed-integer nonlinear optimization by outer approximation—user’s guide, version 1.0, Report, Department of Computer Science, University of Bayreuth (2009)

  37. Lehmann, T., Schittkowski, K., Spickenreuther, T.: BFOUR: a Fortran subroutine for integer optimization by branch-and-bound—user’s guide, Report, Department of Computer Science, University of Bayreuth, Germany (2009)

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  Google Scholar 

  40. Lootsma, F.A.: Performance evaluation of nonlinear optimization methods via multi-criteria analysis and via linear model analysis. In: Powell, M.J.D. (ed.) Nonlinear Optimization, vol. 82. Academic Press, San Diego (1982)

  41. Maratos, N.: Exact penalty function algorithms for finite-dimensional and control optimization problems, Ph.D. thesis, University of London, England (1978)

  42. Nowak, I., Alperin, H., Vigerske, S.: LaGO—an object oriented library for solving MINLPs. In: Bliek, C., Jermann, C., Neumaier, A. (eds.) Global Optimization and Constraint Satisfaction. Lecture Notes in Computer Science, vol. 2861, pp. 32–42. Springer, Berlin (2003)

  43. Powell, M.J.D.: ZQPCVX, A FORTRAN subroutine for convex quadratic programming, Report DAMTP/1983/NA17, University of Cambridge, England (1983)

  44. Quesada I., Grossmann I.E.: An LP/NLP based branch-and-bound algorithm for convex MINLP optimization problems. Comput. Chem. Eng. 16, 937–947 (1992)

    Article  Google Scholar 

  45. Ray T., Sarker R.: Genetic algorithm for solving a gas lift optimization problem. J. Petroleum Sci. Eng. 59, 84–96 (2007)

    Article  Google Scholar 

  46. Saaty T.L.: A scaling method for priorities in hierarchical structures. J. Math. Psychol. 15, 234–281 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  47. Sahinidis, N.V., Tawarmalani, M.: BARON 9.0.4: Global Optimization of Mixed-Integer Nonlinear Programs, User’s Manual (2010). http://www.gams.com/dd/docs/solvers/baron.pdf

  48. Schittkowski, K.: Nonlinear Programming Codes - Information, Tests, Performance. Lecture Notes in Economics and Mathematical Systems, vol. 183. Springer, Berlin (1980)

  49. Schittkowski, K.: QL: A Fortran code for convex quadratic programming—User’s guide, Report, Department of Mathematics, University of Bayreuth, Germany (2003)

  50. Schittkowski, K.: A collection of 100 test problems for nonlinear mixed-integer programming in Fortran—user’s guide, Report, Department of Computer Science, University of Bayreuth (2010)

  51. Schittkowski, K., Yuan, Y.-X.: Sequential quadratic programming methods to appear: Wiley Encyclopedia of Operations Research and Management Science (2010)

  52. Schlueter, M.: Nonlinear Mixed Integer Based Optimization Techniques for Space Applications. Dissertation, School of Mathematics, University of Birmingham (2012)

  53. Tawarmalani M., Sahinidis N.V.: A polyhedral branch-and-cut approach to global optimization. Math. Program. 103, 225–249 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  54. Thomas I., Kröner, A.: Mixed-integer optimization of distillation column tray position in industrial practice. In: Marquardt, W., Pantelides, C. (eds.) 16th European Symposium on Computer Aided Engineering ans 9th International Symposium on Porcess Systems Engineering. Elsevier, Amsterdam, pp. 1015–1020 (2006)

  55. Toint P.L.: A non-monotone trust region algorithm for nonlinear optimization subject to convex constraints. Math. Program. 77(1), 69–94 (1997)

    MathSciNet  MATH  Google Scholar 

  56. Viswanathan J., Grossmann I.E.: A combined penalty function and outer approximation method for MINLP optimization. Comput. Chem. Eng. 14, 769–782 (1990)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  58. Yuan Y.X.: On the convergence of a new trust region algorithm. Numerische Mathematik 70, 515–539 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  59. Yuan Y.X., Sun W.: Optimization Theory and Methods. Springer, Berlin (2006)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Bayreuth, 95440, Bayreuth, Germany

    Oliver Exler & Klaus Schittkowski

  2. Konrad-Zuse-Zentrum (ZIB), Takustr. 7, 14195, Berlin, Germany

    Thomas Lehmann

Authors
  1. Oliver Exler
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Thomas Lehmann
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Klaus Schittkowski
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Klaus Schittkowski.

Additional information

Sponsored by Shell GameChanger, SIEP Rijswijk, under project number 4600003917.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Exler, O., Lehmann, T. & Schittkowski, K. A comparative study of SQP-type algorithms for nonlinear and nonconvex mixed-integer optimization. Math. Prog. Comp. 4, 383–412 (2012). https://doi.org/10.1007/s12532-012-0045-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12532-012-0045-0

Keywords

Mathematics Subject Classification

Search

Navigation