Optimization Letters

, Volume 8, Issue 3, pp 1167–1182 | Cite as

A note on solving MINLP’s using formulation space search

Short Communication

Abstract

In this note we present an approach based on formulation space search to solve mixed-integer nonlinear (zero-one) programming problems. Our approach is an iterative one which adds a single nonlinear inequality constraint of increasing tightness to the original problem. Computational results are presented for our approach on 51 standard benchmark problems taken from MINLPLib. We compare our approach against the Minotaur and minlp_bb nonlinear solvers, as well as against the RECIPE algorithms.

Keywords

Mixed-integer nonlinear (zero-one) programming Formulation space search 

Notes

Acknowledgments

The first author is studying at Brunel with grant support from CONACYT, the Mexican National Council for Science and Technology.

References

  1. 1.
    Belotti, P., Kirches, C., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-integer nonlinear optimization. Preprint ANL/MCS-P3060-1121, Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA (2012)Google Scholar
  2. 2.
    Bonami, P., Kilinc, M., Linderoth, J.: Algorithms and software for convex mixed integer nonlinear programs. In: Lee, J., Leyffer, S. (eds.) Mixed integer nonlinear programming, IMA Volumes in Mathematics and its Applications, vol. 154, pp. 1–39. Springer, New York (2012)CrossRefGoogle Scholar
  3. 3.
    Burer, S., Letchford, A.N.: Non-convex mixed-integer nonlinear programming: a survey. Surv. Oper. Res. Manag. Sci. 17(2), 97–106 (2012)MathSciNetGoogle Scholar
  4. 4.
    Bussieck, M.R., Drud, A.S., Meeraus, A.: MINLPLib - A collection of test models for mixed-integer nonlinear programming. Inf. J. Comput. 15(1), 114–119 (2003)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Bussieck, M.R., Pruessner, A.: Mixed-integer nonlinear programming. SIAG/OPT Views-and-News. 14(1), 19–22 (2003)Google Scholar
  6. 6.
    Bussieck, M.R., Vigerske, S.: MINLP solver software. In: Cochran, J.J., Cox, L.A. Jr., Keskinocak, P., Kharoufeh, J.P., Smith, J.C. (eds.) Wiley Encyclopaedia of Operations Research and Management Science, Wiley, New York, 2011. Available from http://www2.mathematik.hu-berlin.de/~stefan/minlpsoft.pdf last accessed February 5 (2013)
  7. 7.
    Butenko, S., Yezerska, O., Balasundaram, B.: Variable objective search. J. Heuristics. Available at http://dx.doi.org/10.1007/s10732-011-9174-2 (2013) Accessed Feb 15 2013
  8. 8.
    D’Ambrosio, C.: Application-oriented mixed integer non-linear programming. PhD thesis, University of Bologna, Italy, 2009. Available from http://www.lix.polytechnique.fr/~dambrosio/DAmbrosio_Claudia_tesi.pdf last accessed January 18 2013
  9. 9.
    D’Ambrosio, C., Lodi, A.: Mixed integer nonlinear programming tools: a practical overview. 4OR - A Quart. J. Operat. Res. 9(4), 329–349 (2011)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    D’Ambrosio, C., Lodi, A.: Mixed integer nonlinear programming tools: an updated practical overview. Annals of Operations Research. Available at http://dx.doi.org/10.1007/s10479-012-1272-5 (2013) Accessed Feb 15 2013
  11. 11.
    Floudas, C.A.: Nonlinear and mixed-integer optimization: Fundamentals and applications. Oxford University Press, Oxford (1995)MATHGoogle Scholar
  12. 12.
    Floudas, C.A., Gounaris, C.E.: A review of recent advances in global optimization. J. Global Optimiz. 45(1), 3–38 (2009)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Fourer, R., Gay, D.M., Kernighan, B.W.: AMPL A modeling language for mathematical programming. available at http://www.ampl.com/ last accessed January 14 2013
  14. 14.
    Hansen, P., Mladenović, N., Brimberg, J., Perez, J.A.M.: Variable neighborhood search. In: Gendreau, M., Potvin, J.-Y. (eds.) Handbook of Metaheuristics, International series in operations research and management science, vol. 146, pp. 61–86 Springer, Berlin (2010)Google Scholar
  15. 15.
    Hemmecke, R., Koppe, M., Lee, J., Weismantel, J.: Nonlinear integer programming. In: Junger, M., Liebling, T.M., Naddef, D., Nemhauser, G.L., Pulleyblank, W.R., Reinelt, G., Rinaldi, G., Wolsey, L.A. (eds.) 50 years of integer programming 1958–2008, pp. 561–618. Springer, Berlin (2010)Google Scholar
  16. 16.
    Hertz, A., Plumettaz, M., Zufferey, N.: Variable space search for graph coloring. Discret Appl. Math. 156(13), 2551–2560 (2008)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Hertz, A., Plumettaz, M., Zufferey, N.: Corrigendum to “Variable space search for graph coloring” . Discrete Applied Mathematics 157(7), 1335–1336 (2009)Google Scholar
  18. 18.
    Kochetov, Y., Kononova, P., Paschenko, M.: Formulation space search approach for the teacher/class timetabling problem. Yugosl. J. Operat. Res. 18(1), 1–11 (2008)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Lee, J., Leyffer, S. (eds.): Mixed integer nonlinear programming. IMA Volumes in Mathematics and its Applications, vol. 154. Springer, New York (2012)Google Scholar
  20. 20.
    Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A., Munson, T.: Minotaur solver available at http://wiki.mcs.anl.gov/minotaur/index.php/MINOTAUR last accessed January 15 (2013)
  21. 21.
    Leyffer, S., Linderoth, J., Luedtke, J., Miller, A., Munson, T.: Applications and algorithms for mixed integer nonlinear programming. In: Simon, H. (ed.) SCIDAC 2009: Scientific Discovery Through Advanced Computing. Journal of Physics Conference Series 180, 1–5 (2009)Google Scholar
  22. 22.
    Liberti, L., Nannicini, G., Mladenović, N.: A good recipe for solving MINLPs. In: Maniezzo, V., Stützle, T., Voß, S. (eds.) Matheuristics, Annals of Information Systems, Volume 10, pp. 231–244. Springer, Berlin (2010)Google Scholar
  23. 23.
    Liberti, L., Nannicini, G., Mladenović, N.: A recipe for finding good solutions to MINLPs. Math. Program. Comput. 3(4), 349–390 (2011)CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    López, C.O., Beasley, J.E.: A heuristic for the circle packing problem with a variety of containers. Europ. J. Oper. Res. 214(3), 512–525 (2011)CrossRefMATHGoogle Scholar
  25. 25.
    López, C.O., Beasley, J.E.: Packing unequal circles using formulation space search. Comput. Oper. Res. 40(5), 1276–1288 (2013)CrossRefMathSciNetGoogle Scholar
  26. 26.
    MINLP Library, available at http://www.gamsworld.org/minlp/minlplib.htm last accessed January 14 (2013)
  27. 27.
    Mittelmann, H.D.: Performance of commercial and noncommercial optimization software. Presented at INFORMS 2012 Phoenix, Arizona, USA, 2012. Available from http://plato.asu.edu/talks/phoenix.pdf last accessed January 15 (2013)
  28. 28.
    Mladenović, N., Plastria, F., Urošević, D.: Reformulation descent applied to circle packing problems. Comput. Oper. Res. 32(9), 2419–2434 (2005)CrossRefMATHGoogle Scholar
  29. 29.
    Mladenović, N., Plastria, F., Urošević, D.: Formulation space search for circle packing problems. In: “Engineering Stochastic Local Search Algorithms. Designing, Implementing and Analyzing Effective Heuristics”, Proceedings of the International Workshop, SLS 2007, Brussels, Belgium, September 6–8, 2007. Lecture Notes in Computer Science 4638, 212–216 (2007)Google Scholar
  30. 30.
    Pardo, E.G., Mladenović, N., Pantrigo, J.J., Duarte, A.: Variable formulation search for the cutwidth minimization problem. Applied Soft Computing. Available at http://dx.doi.org/10.1016/j.asoc.2013.01.016 (2013) Accessed Feb 15 2013.
  31. 31.
    Vigerske, S.: Private communication (2013)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Mathematical SciencesBrunel UniversityUxbridgeUK

Personalised recommendations