Skip to main content
Springer Nature Link
Log in

Linearization-based algorithms for mixed-integer nonlinear programs with convex continuous relaxation

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

Abstract

We present two linearization-based algorithms for mixed-integer nonlinear programs (MINLPs) having a convex continuous relaxation. The key feature of these algorithms is that, in contrast to most existing linearization-based algorithms for convex MINLPs, they do not require the continuous relaxation to be defined by convex nonlinear functions. For example, these algorithms can solve to global optimality MINLPs with constraints defined by quasiconvex functions. The first algorithm is a slightly modified version of the LP/NLP-based branch-and-bouund \((\text{ LP/NLP-BB })\) algorithm of Quesada and Grossmann, and is closely related to an algorithm recently proposed by Bonami et al. (Math Program 119:331–352, 2009). The second algorithm is a hybrid between this algorithm and nonlinear programming based branch-and-bound. Computational experiments indicate that the modified LP/NLP-BB method has comparable performance to LP/NLP-BB on instances defined by convex functions. Thus, this algorithm has the potential to solve a wider class of MINLP instances without sacrificing performance.

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. Abhishek, K., Leyffer, S., Linderoth, J.: FilMINT: an outer-approximation-based solver for convex mixed-integer nonlinear programs. INFORMS J. Comput. 22(4), 555–567 (2010)

    Article  Google Scholar 

  2. Androulakis, I.P., Maranas, C.D., Floudas, C.A.: \(\alpha \)BB: a global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7, 337–363 (1995)

    Article  Google Scholar 

  3. Bazaraa, M., Sherali, H., Shetty, C.: Nonlinear Programming: Theory and Algorithms. Wiley, New York (2006)

    Book  Google Scholar 

  4. Bertsekas, D., Gallager, R.: Data Networks. Prentice-Hall, Endlewood Cliffs, NJ (1987)

    Google Scholar 

  5. Bertsekas, D., Nedić, A., Ozdaglar, A.: Convex Analysis and Optimization. Athena Scientific, Belmont (2003)

    Google Scholar 

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

    Article  Google Scholar 

  7. Bonami, P., Cornuéjols, G., Lodi, A., Margot, F.: A feasibility pump for mixed integer nonlinear programs. Math. Program. 119, 331–352 (2009)

    Article  Google Scholar 

  8. Bonami, P., Kilinç, M., Linderoth, J.: Algorithms and software for convex mixed integer nonlinear programs. In: Lee, J., Leyffer, S. (eds.) IMA Volumes in Mathematics and its Applications, vol. 154, pp. 1–40. Springer, Berlin (2012)

    Google Scholar 

  9. Boorstyn, R., Frank, H.: Large-scale network topological optimization. IEEE Trans. Commun. 25, 29–47 (1977)

    Article  Google Scholar 

  10. Borchers, B., Mitchell, J.E.: An improved branch and bound algorithm for mixed integer nonlinear programs. Comput. Oper. Res. 21, 359–368 (1994)

    Article  Google Scholar 

  11. Burer, S., Letchford, A.N.: Non-convex mixed-integer nonlinear programming: a survey. Sur. Oper. Res. Manag. Sci. 17, 97–106 (2012)

    Google Scholar 

  12. Bussieck, M.R., Drud, A.: SBB: a new solver for mixed integer nonlinear programming. In: OR 2001, Section: Continuous, Optimization (2001)

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

    Google Scholar 

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

    Article  Google Scholar 

  15. CMU/IBM MINLP Project. http://egon.cheme.cmu.edu/ibm/page.htm

  16. CPLEX Optimization Inc, Incline Village, NV: Using the CPLEX Callable Library, Version 9 (2005)

  17. Dakin, R.J.: A tree search algorithm for mixed integer programming problems. Comput. J. 8, 250–255 (1965)

    Article  Google Scholar 

  18. Dolan, E., Moré, J.: Benchmarking optimization software with performance profiles. Math. Program. 91, 201–212 (2002)

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  22. Fletcher, R., Leyffer, S.: User Manual for FilterSQP (1998). University of Dundee Numerical Analysis, Report NA-181

  23. Flores-Tlacuahuac, A., Biegler, L.T.: Simultaneous mixed-integer dynamic optimization for integrated design and control. Comput. Chem. Eng. 31, 588–600 (2007)

    Article  Google Scholar 

  24. Floudas, C.A.: Deterministic Global Optimization: Theory, Algorithms and Applications. Kluwer, Dordrecht (2000)

    Book  Google Scholar 

  25. Garey, M.R., Johnson, D.S. (eds.): Computers and Interactability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)

    Google Scholar 

  26. Geoffrion, A.: Generalized benders decomposition. J. Optim. Theory Appl. 10(4), 237–260 (1972)

    Article  Google Scholar 

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

    Article  Google Scholar 

  28. Grossmann, I.E., Sargent, R.W.H.: Optimal design of multipurpose chemical plant. Ind. Eng. Chem. Process. Des. Dev. 18, 343–348 (1979)

    Article  Google Scholar 

  29. Grossmann, I.E., Viswanathan, J., Raman, A., Kalvelagen, E.: GAMS/DICOPT: a discrete continuous optimization package. Math. Models Methods Appl. Sci. 11, 649–664 (2001)

    Google Scholar 

  30. Günlük, O., Lee, J., Weismantel, R.: MINLP Strengthening for Separable Convex Quadratic Transportation-Cost ufl. Tech. rep, IBM Research Division (2007)

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

    Google Scholar 

  32. Günlük, O., Linderoth, J.: Perspective reformulation and applications. In: Lee, J., Leyffer, S. (eds.) IMA Volumes in Mathematics and its Applications, vol. 154, pp. 61–92. Springer, Berlin (2012)

    Google Scholar 

  33. Harjunkoski, I., Pörn, R., Westerlund, T.: MINLP: trim-loss problem. In: Floudas, C.A., Pardalos, P.M. (Eds.) Encyclopedia of Optimization, pp. 2190–2198. Springer, Berlin (2009)

  34. Jain, V., Grossmann, I.E.: Cyclic scheduling of continuous parallel-process units with decaying performance. AIChE 44, 1623–1636 (1998)

    Article  Google Scholar 

  35. Laird, C.D., Biegler, L.T., van Bloemen Waanders, B.: A mixed integer approach for obtaining unique solutions in source inversion of drinking water networks. J. Water Resour. Plan. Manag. 132, 242–251 (2006). Sprecial Issue on Drinking Water Distribution Systems Security

    Google Scholar 

  36. Leyffer, S.: User Manual for MINLP-BB. University of Dundee (1998)

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

  38. Mahajan, A., Leyffer, S., Kirches, C.: Solving Mixed-Integer Nonlinear Programs by QP-Diving. Tech. Rep. Preprint ANL/MCS-2004-0112, Argonne National Laboratory, Mathematics and Computer Science Division (March 2012)

  39. Mahajan, A., Leyffer, S., Linderoth, J., Luedtke, J., Munson, T.: MINOTAUR: A Toolkit for Solving Mixed-Integer Nonlinear Optimization. wiki-page (2011). http://wiki.mcs.anl.gov/minotaur/index.php/Main_Page

  40. Mangasarian, O.L.: Nonlinear Programming. In: Classics in Applied Mathematics, 10. Society for Industrial and Applied Mathematics (1994). Originally published 1969 by McGraw-Hill, New York

  41. McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I-convex underestimating problems. Math. Program. 10, 147–175 (1976)

    Article  Google Scholar 

  42. Pörn, R., Westerlund, T.: A cutting plane method for minimizing pseudo-convex functions in the mixed integer case. Comput. Chem. Eng. 24, 2655–2665 (2000)

    Article  Google Scholar 

  43. 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 

  44. 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 

  45. Sawaya, N.: Reformulations, Relaxations and Cutting Planes for Generalized Disjunctive Programming. Ph.D. thesis, Chemical Engineering Department, Carnegie Mellon University (2006)

  46. Still, C., Westerlund, T.: Extended cutting plane algorithm. In: Floudas, C., Pardalos, P. (eds.) Encyclopedia of Optimization, pp. 53–61. Kluwer, Dordrecht (2001)

    Google Scholar 

  47. Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Kluwer, Boston, MA (2002)

    Book  Google Scholar 

  48. Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed integer nonlinear programs: a theoretical and computational study. Math. Program. 99, 563–591 (2004)

    Article  Google Scholar 

  49. 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 

  50. 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 

  51. Wächter, A., Biegler, L.T.: On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear programming. Math. Program. 106, 25–27 (2006)

    Article  Google Scholar 

  52. Westerlund, T., Pettersson, F.: An extended cutting plane method for solving convex minlp problems. Comp. Chem. Eng. 19, 131–136 (1995)

    Article  Google Scholar 

Download references

Acknowledgments

The authors thank Ashutosh Mahajan for his significant help using the MINOTAUR framework to implement our algorithms.

Author information

Authors and Affiliations

  1. Department of Industrial and System Engineering, University of Wisconsin–Madison, Madison, WI, USA

    Mahdi Hamzeei & James Luedtke

Authors
  1. Mahdi Hamzeei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. James Luedtke
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to James Luedtke.

Additional information

This work has been supported by the Office of Advanced Scientific Computing Research (ASCR), Office of Science, U.S. Department of Energy through Grant DE-FG02-08ER25861, and through a contract from Argonne, a U.S. Department of Energy Office of Science laboratory, as part of ASCR’s Applied Mathematics activity.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hamzeei, M., Luedtke, J. Linearization-based algorithms for mixed-integer nonlinear programs with convex continuous relaxation. J Glob Optim 59, 343–365 (2014). https://doi.org/10.1007/s10898-014-0172-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-014-0172-4

Keywords