Skip to main content
Springer Nature Link
Log in

Integrating nonlinear branch-and-bound and outer approximation for convex Mixed Integer Nonlinear Programming

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper, we present a new hybrid algorithm for convex Mixed Integer Nonlinear Programming (MINLP). The proposed hybrid algorithm is an improved version of the classical nonlinear branch-and-bound (BB) procedure, where the enhancements are obtained with the application of the outer approximation algorithm on some nodes of the enumeration tree. The two methods are combined in such a way that each one collaborates to the convergence of the other. Computational experiments with benchmark instances of the MINLP problem show the good performance of the proposed algorithm, which is compared to the outer approximation algorithm, the nonlinear BB algorithm and the hybrid algorithm implemented in the solver Bonmin.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Similar content being viewed by others

References

  1. CMU-IBM open source minlp project

  2. Bonami, P., Biegler, L.T., Conn, A.R., Cornuéjols, G., Grossmann, I.E., Laird, C.D., Lee, J., Lodi, A., Margot, F., Sawaya, N.: An algorithmic framework for convex mixed integer nonlinear programs. Discrete Optim. 5(2), 186–204 (2008)

    Article  Google Scholar 

  3. Bonami, Pierre, Gonçalves, João: Heuristics for convex mixed integer nonlinear programs. Comput. Optim. Appl., pp. 1–19 (2008). doi:10.1007/s10589-010-9350-6

  4. Bonami, P., Kilinç, M., Linderoth, J.: Algorithms and software for convex mixed integer nonlinear programs. Technical report 1664, Computer Sciences Department, University of Wisconsin-Madison (2009)

  5. Bonami, P., Lee, J., Leyffer, S., Wachter, A.: More branch-and-bound experiments in convex nonlinear integer programming. InReport 0 (2011)

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

    Article  Google Scholar 

  7. Christodoulou, M., Costoulakis, C.: Nonlinear mixed integer programming for aircraft collision avoidance in free flight. In: Proceedings of the 12th IEEE Mediterranean Electrotechnical Conference, 2004 MELECON 2004. vol. 1, pp. 327–330, May (2004)

  8. Duran, M., Grossmann, I.: An outer-approximation algorithm for a class of mixed-integer nonlinear programs. Math. Program. 36, 307–339 (1986). doi:10.1007/BF02592064

    Article  Google Scholar 

  9. Geoffrion, A.M.: Generalized benders decomposition. J. Optim. Theory Appl. 10, 237–260 (1972). doi:10.1007/BF00934810

    Article  Google Scholar 

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

  11. Grossmann, I.E., Lee, S.: Generalized convex disjunctive programming: nonlinear convex hull relaxation. Comput. Optim. Appl. 26, 83–100 (2003). doi:10.1023/A:1025154322278

    Article  Google Scholar 

  12. Grossmann, I.E., Ruiz, J.P.: Generalized disjunctive programming: a framework for formulation and alternative algorithms for minlp optimization. In: Lee, J., Leyffer, S. (eds.) Mixed Integer Nonlinear Programming, volume of The IMA Volumes in Mathematics and Its Applications, pp. 93–115. Springer, New York (2012). doi:10.1007/978-1-4614-1927-3_4

    Chapter  Google Scholar 

  13. Gupta, O.K., Ravindran, A.: Branch and bound experiments in convex nonlinear integer programming. Manag. Sci. 31(12), 1533–1546 (1985)

    Article  Google Scholar 

  14. Land, A.H., Doig, A.G.: An automatic method of solving discrete programming problems. Econometrica 28(3), 497–520 (1960)

    Article  Google Scholar 

  15. Lee, S., Grossmann, I.E.: New algorithms for nonlinear generalized disjunctive programming. Comput. Chem. Eng. 24(9–10), 2125–2141 (2000)

    Article  Google Scholar 

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

    Article  Google Scholar 

  17. Leyffer, S., Linderoth, J., Luedtke, J., Miller, A., Munson, T.: Applications and algorithms for mixed integer nonlinear programming. J. Phys. Conf. Ser. 180(1), 012014 (2009)

    Article  Google Scholar 

  18. Melo, W.: Agoritmos para programação não linear inteira mista (Algorithms for Mixed Integer Nonlinear Programming). Master thesis, COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, RJ, Brasil, (2012) Printed in Portuguese

  19. Murray, W., Ng, K.-M.: An algorithm for nonlinear optimization problems with binary variables. Comput. Optim. Appl. 47, 257–288 (2010). doi:10.1007/s10589-008-9218-1

    Article  Google Scholar 

  20. Quesada, I., Grossmann, I.E.: An lp/nlp based branch and bound algorithm for convex minlp optimization problems. Comput .Chem. Eng., 16(10–11), 937–947, (1992). An International Journal of Computer Applications in Chemical Engineering

  21. Raman, R., Grossmann, I.E.: Modelling and computational techniques for logic based integer programming. Comput. Chem. Eng., 18(7), 563–578 (1994). An International Journal of Computer Applications in Chemical Engineering

  22. Still, C., Westerlund, T.: Solving convex minlp optimization problems using a sequential cutting plane algorithm. Comput. Optim. Appl. 34, 63–83 (2006). doi:10.1007/s10589-005-3076-x

    Article  Google Scholar 

  23. Stubbs, R.A., Mehrotra, S.: A branch-and-cut method for 0–1 mixed convex programming. Math. Program. 86, 515–532 (1999). doi:10.1007/s101070050103

    Article  Google Scholar 

  24. Westerlund, T., Pettersson, F.: An extended cutting plane method for solving convex minlp problems. Comput. Chem. Eng. 19(Supplement 10, 131–136 (1995). European Symposium on Computer Aided Process Engineering

  25. You, F., Grossmann, I.E.: Mixed-integer nonlinear programming models and algorithms for large-scale supply chain design with stochastic inventory management. Ind. Eng. Chem. Res. 47(20), 7802–7817 (2008)

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank the anonymous referees, whose comments helped to improve the paper, and CAPES and CNPq for the financial support.

Author information

Authors and Affiliations

  1. Institute for Graduate Studies and Research in Engineering (COPPE), Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Wendel Melo

  2. Institute of Mathematics and COPPE, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

    Marcia Fampa

  3. Department of Industrial Engineering, Pontifical Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil

    Fernanda Raupp

  4. National Laboratory for Scientific Computing, Petrópolis, Brazil

    Fernanda Raupp

Authors
  1. Wendel Melo
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Marcia Fampa
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Fernanda Raupp
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Wendel Melo.

Appendices

Appendix A

1.1 Information about the test problems

Table 1 describes the characteristics of the selected test-instances. The columns of the table provide for each instance: name, objective type, number of continuous variables (\(n_x\)), number of integer (binary) variables (\(n_y\)), number of nonlinear constraints (\(m_{nl}\)), number of quadratic constraints (\(m_q\)), number of linear constraints (\(m_l\)), number of nonzeros in objective function quadratic matrix (\(\#Q\)), number of nonzeros in the Jacobian of the nonlinear constraints (excluding linear and quadratic ones) (\(\#{\nabla g}\)), number of nonzeros in the nonlinear Hessian of the Lagrangian (excluding quadratic expressions) (\(\#{\nabla ^2 H}\)).

Appendix B

1.1 Detailed computational results

Table 2 shows the computational results of OA, BB, our hybrid approach, and Bonmin’s hybrid approach for the 50 test problems selected. The columns on the table present, for each approach: final status (St), best solution (Sol), and CPU time in seconds (Time). The possible values for the final status (St) are: Optimal Solution (OS), Maximum Time Achieved (MT), and Solver Error (SE). Maximum Time was set at 14,400 s (4 h).

Table 2 Computational results for OA, BB, our Hybrid and Bonmin
Full size table

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Melo, W., Fampa, M. & Raupp, F. Integrating nonlinear branch-and-bound and outer approximation for convex Mixed Integer Nonlinear Programming. J Glob Optim 60, 373–389 (2014). https://doi.org/10.1007/s10898-014-0217-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-014-0217-8

Keywords