Skip to main content
Log in

Proximity Benders: a decomposition heuristic for stochastic programs

  • Published:
Journal of Heuristics Aims and scope Submit manuscript

Abstract

In this paper we present a heuristic approach to two-stage mixed-integer linear stochastic programming models with continuous second stage variables. A common solution approach for these models is Benders decomposition, in which a sequence of (possibly infeasible) solutions is generated, until an optimal solution is eventually found and the method terminates. As convergence may require a large amount of computing time for hard instances, the method may be unsatisfactory from a heuristic point of view. Proximity search is a recently-proposed heuristic paradigm in which the problem at hand is modified and iteratively solved with the aim of producing a sequence of improving feasible solutions. As such, proximity search and Benders decomposition naturally complement each other, in particular when the emphasis is on seeking high-quality, but not necessarily optimal, solutions. In this paper, we investigate the use of proximity search as a tactical tool to drive Benders decomposition, and computationally evaluate its performance as a heuristic on instances of different stochastic programming problems.

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

Access this article

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

  • Achterberg, T., Berthold, T., Hendel, G.: Rounding and propagation heuristics for mixed integer programming. In: Operation Research Proceedings 2011, pp. 71–76. Springer, Berlin (2012)

  • Beasley, J.: Or-library: Distributing test problems by electronic mail. J. Oper. Res. Soc. 41, 1069–1072 (1990)

    Article  Google Scholar 

  • Benders, J.: Partitioning procedures for solving mixed-variables programming problems. Numer. Math. 4(1), 238–252 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  • Berthold, T.: Measuring the impact of primal heuristics. Oper. Res. Lett. 41(6), 611–614 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  • Bodur, M., Dash, S., Günlück, O., Luedtke, J.: Strengthened Benders cuts for stochastic integer programs with continuous recourse. Technical Report. Optimization Online 2014-03-4263 (2014)

  • Crainic, T., Fu, X., Gendreau, M., Rei, W., Wallace, S.: Progressive hedging-based metaheuristics for stochastic network design. Networks 58(2), 114–124 (2011)

    MathSciNet  MATH  Google Scholar 

  • Crainic, T., Hewitt, M., Rei, W.: Partial decomposition strategies for two-stage stochastic integer programs. Technical Report 13, CIRRELT (2014)

  • Fischetti, M., Glover, F., Lodi, A.: The feasibility pump. Math. Program. 104(1), 91–104 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  • Fischetti, M., Lodi, A.: Local branching. Math. Program. 98, 23–47 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  • Fischetti, M., Monaci, M.: Proximity search for 0–1 mixed-integer convex programming. J. Heuristics 20(6), 709–731 (2014)

    Article  Google Scholar 

  • Fischetti, M., Salvagnin, D., Zanette, A.: A note on the selection of Benders cuts. Math. Program. 124(1–2), 175–182 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  • Louveaux, L.: Discrete stochastic location models. Ann. Oper. Res. 6, 23–34 (1986)

    Article  Google Scholar 

  • Pan, F., Morton, D.: Minimizing a stochastic maximum-reliability path. Networks 52(3), 111–119 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  • Rei, W., Cordeau, J.-F., Gendreau, M., Soriano, P.: Accelerating Benders decomposition by local branching. INFORMS J. Comput. 21(2), 333–345 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Shaw, P.: Using constraint programming and local search methods to solve vehicle routing problems. In: Maher M., Puget J.-F. (eds.) Principles and Practice of Constraint Programming CP98. Lecture Notes in Computer Science, vol. 1520, pp. 417–431. Springer, Berlin, Heidelberg (1998)

Download references

Acknowledgments

This research of Fischetti and Monaci was supported by the University of Padova (Progetto di Ateneo “Exploiting randomness in Mixed Integer Linear Programming”), and by MiUR, Italy (PRIN project “Mixed-Integer Nonlinear Optimization: Approaches and Applications”). We thank Merve Bodur and Mike Hewitt for their support and for providing the data used in the computational analysis. Thanks are also due to three anonymous referees for their helpful comments leading to an improved presentation.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Matteo Fischetti.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Boland, N., Fischetti, M., Monaci, M. et al. Proximity Benders: a decomposition heuristic for stochastic programs. J Heuristics 22, 181–198 (2016). https://doi.org/10.1007/s10732-015-9306-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10732-015-9306-1

Keywords

Navigation