Skip to main content
SpringerLink
Log in

Branch-and-price for a class of nonconvex mixed-integer nonlinear programs

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

Abstract

This work attempts to combine the strengths of two major technologies that have matured over the last three decades: global mixed-integer nonlinear optimization and branch-and-price. We consider a class of generally nonconvex mixed-integer nonlinear programs (MINLPs) with linear complicating constraints and integer linking variables. If the complicating constraints are removed, the problem becomes easy to solve, e.g. due to decomposable structure. Integrality of the linking variables allows us to apply a discretization approach to derive a Dantzig-Wolfe reformulation and solve the problem to global optimality using branch-andprice. It is a remarkably simple idea; but to our surprise, it has barely found any application in the literature. In this work, we show that many relevant problems directly fall or can be reformulated into this class of MINLPs. We present the branch-and-price algorithm and demonstrate its effectiveness (and sometimes ineffectiveness) in an extensive computational study considering multiple large-scale problems of practical relevance, showing that, in many cases, orders-of-magnitude reductions in solution time can be achieved.

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

Similar content being viewed by others

References

  1. Lübbecke, M.E., Desrosiers, J.: Selected topics in column generation. Oper. Res. 53(6), 1007–1023 (2005). https://doi.org/10.1287/opre.1050.0234

    Article  MathSciNet  MATH  Google Scholar 

  2. Singh, K.J., Philpott, A.B., Wood, R.K.: Dantzig–Wolfe decomposition for solving multistage stochastic capacity-planning problems. Oper. Res. 57(5), 1271–1286 (2009). https://doi.org/10.1287/opre.1080.0678

    Article  MathSciNet  MATH  Google Scholar 

  3. Nowak, I., Breitfeld, N., Hendrix, E.M., Njacheun-Njanzoua, G.: Decomposition-based inner- and outer-refinement algorithms for global optimization. J. Glob. Optim. 72(2), 305–321 (2018). https://doi.org/10.1007/s10898-018-0633-2

    Article  MathSciNet  MATH  Google Scholar 

  4. Dantzig, G., Fulkerson, R., Johnson, S.: Solution of a large-scale traveling-salesman problem. Oper. Res. 2(4), 393–410 (1954)

    MathSciNet  MATH  Google Scholar 

  5. Gomory, R.E.: Outline of an algorithm for integer solutions to linear programs. Bull. Am. Math. Soc. 64(5), 275–278 (1958)

    Article  MathSciNet  Google Scholar 

  6. Jünger, M., Liebling, T., Naddef, D., Nemhauser, G.L., Pulleyblank, W., Reinelt, G., Rinaldi, G., Wolsey, L.A.: 50 Years of Integer Programming 1958–2008: From the Early Years to the State-of-the-Art. Springer (2009)

  7. Grossmann, I.E., Sargent, R.W.: Optimum design of multipurpose chemical plants. Ind. Eng. Chem. Process Des. Dev. 18(2), 343–348 (1979). https://doi.org/10.1021/i260070a031

    Article  Google Scholar 

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

    Article  Google Scholar 

  9. Stubbs, R.A., Mehrotra, S.: A branch-and-cut method for 0–1 mixed convex programming. Math. Progr., Ser. B 86(3), 515–532 (1999). https://doi.org/10.1007/s101070050103

    Article  MathSciNet  MATH  Google Scholar 

  10. Geoffrion, A.M.: Generalized benders decomposition. J. Optim. Theory Appl. 10(4), 237–260 (1972). https://doi.org/10.1097/ACI.0000000000000254

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  13. 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). https://doi.org/10.1016/0098-1354(92)80028-8

    Article  Google Scholar 

  14. Westerlund, T., Pettersson, F.: An extended cutting plane method for solving convex MINLP problems. Comput. Chem. Eng. 19(Suppl. 1), 131–136 (1995). https://doi.org/10.1016/0098-1354(95)87027-X

    Article  Google Scholar 

  15. Kronqvist, J., Lundell, A., Westerlund, T.: The extended supporting hyperplane algorithm for convex mixed-integer nonlinear programming. J. Glob. Optim. 64(2), 249–272 (2016). https://doi.org/10.1007/s10898-015-0322-3

    Article  MathSciNet  MATH  Google Scholar 

  16. Grossmann, I.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3(3), 227–252 (2002). https://doi.org/10.1023/A:1021039126272

    Article  MathSciNet  MATH  Google Scholar 

  17. Bonami, P., Kilinç, M., Linderoth, J.: Algorithms and software for convex mixed integer nonlinear programs. In: Mixed Integer Nonlinear Programming, pp. 1–39. Springer (2012). https://doi.org/10.1007/978-1-4614-1927-3

  18. Kronqvist, J., Bernal, D.E., Lundell, A., Grossmann, I.E.: A Review and Comparison of Solvers for Convex MINLP, vol. 20. Springer (2019). https://doi.org/10.1007/s11081-018-9411-8

  19. McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: part I—convex underestimating problems. Math. Progr. 10(1), 147–175 (1976). https://doi.org/10.1007/BF01580665

    Article  MATH  Google Scholar 

  20. Ryoo, H.S., Sahinidis, N.V.: A branch-and-reduce approach to global optimization. J. Glob. Optim. 8(2), 107–138 (1996). https://doi.org/10.1007/bf00138689

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Androulakis, I.P., Maranas, C.D., Floudas, C.A.: \(\alpha \)BB: A global optimization method for general constrained nonconvex problems. J. Glob. Optim. 7(4), 337–363 (1995). https://doi.org/10.1007/BF01099647

    Article  MathSciNet  MATH  Google Scholar 

  23. Adjiman, C.S., Androulakis, I.P., Floudas, C.A.: Global optimization of mixed-integer nonlinear problems. AIChE J. 46(9), 1769–1797 (2000)

    Article  Google Scholar 

  24. Kılınç, M.R., Sahinidis, N.V.: Exploiting integrality in the global optimization of mixed-integer nonlinear programming problems with BARON. Optim. Methods Softw. 33(3), 540–562 (2018). https://doi.org/10.1080/10556788.2017.1350178

    Article  MathSciNet  MATH  Google Scholar 

  25. Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tighteningtechniques for non-convex MINLP. Optim. Methods Softw. 24(4–5), 597–634 (2009). https://doi.org/10.1080/10556780903087124

    Article  MathSciNet  MATH  Google Scholar 

  26. Lin, Y., Schrage, L.: The global solver in the LINDO API. Optim. Methods Softw. 24(4–5), 657–668 (2009). https://doi.org/10.1080/10556780902753221

    Article  MathSciNet  MATH  Google Scholar 

  27. Misener, R., Floudas, C.A.: ANTIGONE: algorithms for continuous/integer global optimization of nonlinear equations. J. Glob. Optim. 59(2–3), 503–526 (2014). https://doi.org/10.1007/s10898-014-0166-2

  28. Vigerske, S., Gleixner, A.: SCIP: global optimization of mixed-integer nonlinear programs in a branch-and-cut framework. Optim. Methods Softw. 33(3), 563–593 (2018). https://doi.org/10.1080/10556788.2017.1335312

    Article  MathSciNet  MATH  Google Scholar 

  29. Burer, S., Letchford, A.N.: Non-convex mixed-integer nonlinear programming: a survey. Surv. Oper. Res. Manag. Sci. 17(2), 97–106 (2012). https://doi.org/10.1016/j.sorms.2012.08.001

    Article  MathSciNet  Google Scholar 

  30. Boukouvala, F., Misener, R., Floudas, C.A.: Global optimization advances in mixed-integer nonlinear programming, MINLP, and constrained derivative-free optimization. CDFO. Eur. J. Oper. Res. 252(3), 701–727 (2016). https://doi.org/10.1016/j.ejor.2015.12.018

    Article  MathSciNet  MATH  Google Scholar 

  31. Guignard, M.: Lagrangean relaxation. Top 11(2), 151–200 (2003). https://doi.org/10.1007/BF02579036

    Article  MathSciNet  MATH  Google Scholar 

  32. Watson, J.P., Woodruff, D.L.: Progressive hedging innovations for a class of stochastic mixed-integer resource allocation problems. Comput. Manag. Sci. 8(4), 355–370 (2011). https://doi.org/10.1007/s10287-010-0125-4

    Article  MathSciNet  MATH  Google Scholar 

  33. Lotero, I., Trespalacios, F., Grossmann, I.E., Papageorgiou, D.J., Cheon, M.S.: An MILP-MINLP decomposition method for the global optimization of a source based model of the multiperiod blending problem. Comput. Chem. Eng. 87, 13–35 (2016). https://doi.org/10.1016/j.compchemeng.2015.12.017

    Article  Google Scholar 

  34. Lara, C.L., Trespalacios, F., Grossmann, I.E.: Global optimization algorithm for capacitated multi-facility continuous location-allocation problems. J. Glob. Optim. 71(4), 871–889 (2018). https://doi.org/10.1007/s10898-018-0621-6

    Article  MathSciNet  MATH  Google Scholar 

  35. Elsido, C., Martelli, E., Grossmann, I.E.: A bilevel decomposition method for the simultaneous heat integration and synthesis of steam/organic Rankine cycles. Comput. Chem. Eng. 128, 228–245 (2019). https://doi.org/10.1016/j.compchemeng.2019.05.041

    Article  Google Scholar 

  36. Li, X., Tomasgard, A., Barton, P.I.: Nonconvex generalized benders decomposition for stochastic separable mixed-integer nonlinear programs. J. Optim. Theory Appl. 151(3), 425–454 (2011). https://doi.org/10.1007/s10957-011-9888-1

    Article  MathSciNet  MATH  Google Scholar 

  37. Cao, Y., Zavala, V.M.: A scalable global optimization algorithm for stochastic nonlinear programs. J. Glob. Optim. 75(2), 393–416 (2019). https://doi.org/10.1007/s10898-019-00769-y

    Article  MathSciNet  MATH  Google Scholar 

  38. Li, C., Grossmann, I.E.: A generalized Benders decomposition-based branch and cut algorithm for two-stage stochastic programs with nonconvex constraints and mixed-binary first and second stage variables. J. Glob. Optim. 75(2), 247–272 (2019). https://doi.org/10.1007/s10898-019-00816-8

    Article  MathSciNet  MATH  Google Scholar 

  39. Rebennack, S., Kallrath, J., Pardalos, P.M.: Column enumeration based decomposition techniques for a class of non-convex MINLP problems. J. Glob. Optim. 43(2–3), 277–297 (2009). https://doi.org/10.1007/s10898-007-9271-9

    Article  MathSciNet  MATH  Google Scholar 

  40. Dantzig, G.B., Wolfe, P.: Decomposition principle for linear programs. Oper. Res. 8(1), 101–111 (1960)

    Article  Google Scholar 

  41. Desrosiers, J., Soumis, F., Desrochers, M.: Routing with time windows by column generation. Networks 14(4), 545–565 (1984). https://doi.org/10.1002/net.3230140406

    Article  MATH  Google Scholar 

  42. Desrochers, M., Desrosiers, J., Solomon, M.: A new optimization algorithm for the vehicle routing problem with time windows. Oper. Res. 40(2), 342–354 (1992)

    Article  MathSciNet  Google Scholar 

  43. Desaulniers, G., Desrosiers, J., Solomon, M.M.: Accelerating strategies in column generation methods for vehicle routing and crew scheduling problems. In: Essays and Surveys in Metaheuristics, pp. 309–324. Springer (2002)

  44. Desrochers, M., Soumis, F.: A column generation approach to the urban transit crew scheduling problem. Transp. Sci. 23(1), 1–13 (1989). https://doi.org/10.1287/trsc.23.1.1

    Article  MATH  Google Scholar 

  45. Stojković, M., Soumis, F., Desrosiers, J.: The operational airline crew scheduling problem. Transp. Sci. 32(3), 232–245 (1998). https://doi.org/10.1287/trsc.1090.0306

    Article  MATH  Google Scholar 

  46. Ioachim, I., Desrosiers, J., Soumis, F., Bélanger, N.: Fleet assignment and routing with schedule synchronization constraints. Eur. J. Oper. Res. 119(1), 75–90 (1999). https://doi.org/10.1016/S0377-2217(98)00343-9

    Article  MATH  Google Scholar 

  47. Bélanger, N., Desaulniers, G., Soumis, F., Desrosiers, J.: Periodic airline fleet assignment with time windows, spacing constraints, and time dependent revenues. Eur. J. Oper. Res. 175(3), 1754–1766 (2006). https://doi.org/10.1016/j.ejor.2004.04.051

    Article  MathSciNet  MATH  Google Scholar 

  48. Barnhart, C., Johnson, E.L., Nemhauser, G.L., Savelsbergh, M.W.P., Vance, P.H.: Branch-and-price: column generation for solving huge integer programs. Oper. Res. 46(3), 316–329 (1998). https://doi.org/10.1287/opre.46.3.316

    Article  MathSciNet  MATH  Google Scholar 

  49. Vanderbeck, F.: On Dantzig-Wolfe decomposition in integer programming and ways to perform branching in a branch-and-price algorithm. Oper. Res. 48(1), 111–128 (2000). https://doi.org/10.1287/opre.48.1.111.12453

    Article  MathSciNet  MATH  Google Scholar 

  50. Wolsey, L.A.: Integer Programming. Wiley (1998)

  51. Lubin, M., Dunning, I.: Computing in operations research using Julia. INFORMS J. Comput. 27, 237–248 (2015)

    Article  MathSciNet  Google Scholar 

  52. Gilmore, P.C., Gomory, R.E.: A linear programming approach to the cutting-stock problem. Oper. Res. 9, 849–859 (1961)

    Article  MathSciNet  Google Scholar 

  53. Rajagopalan, S., Yu, H.L.: Capacity planning with congestion effects. Eur. J. Oper. Res. 134(2), 365–377 (2001). https://doi.org/10.1016/S0377-2217(00)00254-X

    Article  MATH  Google Scholar 

  54. Karuppiah, R., Grossmann, I.E.: Global optimization of multiscenario mixed integer nonlinear programming models arising in the synthesis of integrated water networks under uncertainty. Comput. Chem. Eng. 32, 145–160 (2008). https://doi.org/10.1016/j.compchemeng.2007.03.007

    Article  Google Scholar 

Download references

Acknowledgements

The authors gratefully acknowledge financial support from the University of Minnesota and the Minnesota Supercomputing Institute (MSI) at the University of Minnesota for providing resources that contributed to the research results reported within this paper. We also thank Angela Flores-Quiroz for insightful discussions on our work.

Author information

Authors and Affiliations

  1. Department of Chemical Engineering and Materials Science, University of Minnesota, Twin Cities, Minneapolis, MN, 55455, USA

    Andrew Allman & Qi Zhang

Authors
  1. Andrew Allman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qi Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qi Zhang.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Allman, A., Zhang, Q. Branch-and-price for a class of nonconvex mixed-integer nonlinear programs. J Glob Optim 81, 861–880 (2021). https://doi.org/10.1007/s10898-021-01027-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-021-01027-w

Keywords

Search

Navigation