Skip to main content

Advertisement

SpringerLink
Log in

Strong-branching inequalities for convex mixed integer nonlinear programs

  • Published:
Computational Optimization and Applications Aims and scope Submit manuscript

Abstract

Strong branching is an effective branching technique that can significantly reduce the size of the branch-and-bound tree for solving mixed integer nonlinear programming (MINLP) problems. The focus of this paper is to demonstrate how to effectively use “discarded” information from strong branching to strengthen relaxations of MINLP problems. Valid inequalities such as branching-based linearizations, various forms of disjunctive inequalities, and mixing-type inequalities are all discussed. The inequalities span a spectrum from those that require almost no extra effort to compute to those that require the solution of an additional linear program. In the end, we perform an extensive computational study to measure the impact of each of our proposed techniques. Computational results reveal that existing algorithms can be significantly improved by leveraging the information generated as a byproduct of strong branching in the form of valid inequalities.

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

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

Similar content being viewed by others

References

  1. Abhishek, K., Leyffer, S., Linderoth, J.T.: FilMINT: an outer-approximation-based solver for nonlinear mixed integer programs. INFORMS J. Comput. 22, 555–567 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  2. Achterberg, T.: Constraint Integer Programming. Ph.D. Thesis, Technischen Universtät Berlin (2007)

  3. Achterberg, T., Koch, T., Martin, A.: Branching rules revisited. Oper. Res. Lett. 33, 42–54 (2004)

    Article  MathSciNet  Google Scholar 

  4. Applegate, D., Bixby, R., Cook, W., Chvátal, V.: The Traveling Salesman Problem, A Computational Study. Princeton University Press, Princeton (2006)

    MATH  Google Scholar 

  5. Atamtürk, A., Nemhauser, G., Savelsbergh, M.W.P.: Conflict graphs in solving integer programming problems. Eur. J. Oper. Res. 121, 40–55 (2000)

    Article  MATH  Google Scholar 

  6. Balas, E.: A modified lift-and-project procedure. Math. Program. 79, 19–31 (1997)

    MATH  MathSciNet  Google Scholar 

  7. Balas, E.: Disjunctive programming: properties of the convex hull of feasible points. Discr. Appl. Math. 89(1–3), 3–44 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Balas, E., Ceria, S., Cornuéjols, G.: A lift-and-project cutting plane algorithm for mixed 0–1 programs. Math. Program. 58, 295–324 (1993)

    Article  MATH  Google Scholar 

  9. Balas, E., Ceria, S., Cornuéjols, G.: Mixed 0–1 programming by lift-and-project in a branch-and-cut framework. Manag. Sci. 42, 1229–1246 (1996)

    Article  MATH  Google Scholar 

  10. Balas, E., Jeroslow, R.G.: Strengthening cuts for mixed integer programs. Eur. J. Oper. Res. 4, 224–234 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  11. Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-integer nonlinear optimization. Acta Numer. 22, 1–131 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bonami, P., Kılınç, M., Linderoth, J.: Algorithms and software for solving convex mixed integer nonlinear programs. IMA Vol. Math. Appl. 54, 1–40 (2012)

    Article  Google Scholar 

  13. Bonami, P., Lee, J., Leyffer, S., Wächter, A.: More branch-and-bound experiments in convex nonlinear integer programming. Preprint ANL/MCS-P1949-0911, Argonne National Laboratory, Mathematics and Computer Science Division, September (2011)

  14. Boorstyn, R.R., Frank, H.: Large-scale network topological optimization. IEEE Trans. Commun. 25, 29–47 (1997)

  15. Bussieck, M.R., Drud, A.S., Meeraus, A.: MINLPLib - a collection of test models for mixed-integer nonlinear programming. INFORMS J. Comput. 15(1), 114–119 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  16. Castillo, I., Westerlund, J., Emet, S., Westerlund, T.: Optimization of block layout deisgn problems with unequal areas: a comparison of MILP and MINLP optimization methods. Comput. Chem. Eng. 30, 54–69 (2005)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Elhedhli, S.: Service system design with immobile servers, stochastic demand, and congestion. Manuf. Serv. Oper. Manag. 8(1), 92–97 (2006)

    Google Scholar 

  20. Fischetti, M., Lodi, A., Tramontani, A.: On the separation of disjunctive cuts. Math. Programm. 128, 205–230 (2011)

  21. Fletcher, R., Leyffer, S.: User manual for filterSQP. University of Dundee Numerical Analysis Report NA-181 (1998)

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

    Article  MATH  MathSciNet  Google Scholar 

  23. Günlük, O., Lee, J., Weismantel, R.: MINLP strengthening for separaable convex quadratic transportation-cost ufl. Technical Report RC24213 (W0703–042), IBM Research Division, March (2007)

  24. Günlük, O., Pochet, Y.: Mixing mixed-integer inequalities. Math. Program. 90(3), 429–457 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  25. Harjunkoski, I., Pörn, R., Westerlund, T.: MINLP: Trim-loss problem. In: Floudas, Christodoulos A., Pardalos, Panos M. (eds.) Encyclopedia of Optimization, pp. 2190–2198. Springer, New York (2009)

    Chapter  Google Scholar 

  26. Kılınç, M.: Disjunctive Cutting Planes and Algorithms for Convex Mixed Integer Nonlinear Programming. Ph.D. Thesis, University of Wisconsin-Madison (2011)

  27. Leyffer, S.: MacMINLP: Test Problems for Mixed Integer Nonlinear Programming, (2003). http://www.mcs.anl.gov/~leyffer/macminlp

  28. Linderoth, J.T., Savelsbergh, M.W.P.: A computational study of search strategies in mixed integer programming. INFORMS J. Comput. 11, 173–187 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  29. Nemhauser, G.L., Savelsbergh, M.W.P., Sigismondi, G.C.: MINTO, a Mixed INTeger Optimizer. Oper. Res. Lett. 15, 47–58 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  30. Pochet, Y., Wolsey, L.: Lot sizing with constant batches: formulation and valid inequalities. Math. Oper. Res. 18, 767–785 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  31. 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 

  32. Ravemark, D.E., Rippin, D.W.T.: Optimal design of a multi-product batch plant. Comput. Chem. Eng. 22(1–2), 177–183 (1998)

    Article  Google Scholar 

  33. Sawaya, N.: Reformulations, relaxations and cutting planes for generalized disjunctive programming. Ph.D. Thesis, Chemical Engineering Department, Carnegie Mellon University (2006)

  34. Sawaya, N., Laird, C.D., Biegler, L.T., Bonami, P., Conn, A.R., Cornuéjols, G., Grossmann, I.E., Lee, J., Lodi, A., Margot, F., Wächter, A.: CMU-IBM open source MINLP project test set, (2006). http://egon.cheme.cmu.edu/ibm/page.htm

  35. Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: extended formulations. Math. Program. Ser. B 124, 383–411 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  37. Türkay, M., Grossmann, I.E.: Logic-based MINLP algorithms for the optimal synthesis of process networks. Comput. Chem. Eng. 20(8), 959–978 (1996)

    Article  Google Scholar 

  38. Vecchietti, A., Grossmann, I.E.: LOGMIP: a disjunctive 0–1 non-linear optimizer for process system models. Comput. Chem. Eng. 23(4–5), 555–565 (1999)

    Article  Google Scholar 

Download references

Acknowledgments

The authors would like to thank two anonymous referees for their useful comments and patience. This research was supported in part by the Office of Advanced Scientific Computing Research, Office of Science, U.S. Department of Energy under Grant DE-FG02-08ER25861 and by the U.S. National Science Foundation under Grant CCF-0830153.

Author information

Authors and Affiliations

  1. Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Mustafa Kılınç

  2. Department of Industrial and Systems Engineering, University of Wisconsin-Madison, Madison, WI, 53706, USA

    Jeff Linderoth & James Luedtke

  3. United Parcel Service, 30328, Atlanta, GA, USA

    Andrew Miller

Authors
  1. Mustafa Kılınç
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeff Linderoth
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. James Luedtke
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Andrew Miller
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jeff Linderoth.

Electronic supplementary material

Below is the link to the electronic supplementary material.

ESM 1 (PDF 103 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kılınç, M., Linderoth, J., Luedtke, J. et al. Strong-branching inequalities for convex mixed integer nonlinear programs. Comput Optim Appl 59, 639–665 (2014). https://doi.org/10.1007/s10589-014-9690-8

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10589-014-9690-8

Keywords

Search

Navigation