Skip to main content
SpringerLink
Log in

A finite branch-and-bound algorithm for two-stage stochastic integer programs

  • Published:
Mathematical Programming Submit manuscript

Abstract.

This paper addresses a general class of two-stage stochastic programs with integer recourse and discrete distributions. We exploit the structure of the value function of the second-stage integer problem to develop a novel global optimization algorithm. The proposed scheme departs from those in the current literature in that it avoids explicit enumeration of the search space while guaranteeing finite termination. Computational experiments on standard test problems indicate superior performance of the proposed algorithm in comparison to those in the existing literature.

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

Similar content being viewed by others

References

  1. Ahmed, S.: Strategic Planning under Uncertainty: Stochastic Integer Programming Approaches. Ph.D. Thesis, University of Illinois, Urbana, IL, 2000

  2. Birge, J.R., Louveaux, F.V.: Introduction to Stochastic Programming. Springer, New York, NY, 1997

  3. Blair, C.E., Jeroslow, R.G.: The value function of an integer program. Math. Program. 23, 237–273 (1982)

    MathSciNet  MATH  Google Scholar 

  4. Blair, C.E., Jeroslow, R.G.: Constructive characterizations of the value-function of a mixed-integer program I. Disc. Appl. Math. 9, 217–233 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carøe, C.C.: Decomposition in stochastic integer programming. PhD thesis, University of Copenhagen, 1998

  6. Carøe, C.C., Schultz, R.: Dual decomposition in stochastic integer programming. Oper. Res. Lett. 24, 37–45 (1999)

    Article  Google Scholar 

  7. Carøe, C.C., Tind, J.: L-shaped decomposition of two-stage stochastic programs with integer recourse. Math. Program. 83, 451–464 (1998)

    Article  Google Scholar 

  8. Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches. Springer-Verlag, Berlin, 3rd edition, 1996

  9. Jorjani, S., Scott, C.H., Woodruff, D.L.: Selection of an optimal subset of sizes. Technical report, University of California, Davis, CA, 1995

  10. Kall, P., Wallace, S.W.: Stochastic Programming. John Wiley and Sons, Chichester, England, 1994

  11. Klein Haneveld, W.K., Stougie, L., van der Vlerk, M.H.: On the convex hull of the simple integer recourse objective function. Ann. Oper. Res. 56, 209–224 (1995)

    MATH  Google Scholar 

  12. Klein Haneveld, W.K., Stougie, L., van der Vlerk, M.H.: An algorithm for the construction of convex hulls in simple integer recourse programming. Ann. Oper. Res. 64, 67–81 (1996)

    MATH  Google Scholar 

  13. Klein Haneveld, W.K., van der Vlerk, M.H.: Stochastic integer programming: General models and algorithms. Ann. Oper. Res. 85, 39–57 (1999)

    Article  MATH  Google Scholar 

  14. Laporte, G., Louveaux, F.V.: The integer L-shaped method for stochastic integer programs with complete recourse. Oper. Res. Lett. 13, 133–142 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  15. Norkin, V.I., Ermoliev, Y.M., Ruszczyński, A.: On optimal allocation of indivisibles under uncertainty. Oper. Res. 46(3), 381–395 (1998)

    MATH  Google Scholar 

  16. Rockafellar, R.T.: Convex Analysis. Princeton University Press, New Jersey, 1970

  17. Rockafellar, R.T., Wets, R.J.-B.: Scenarios and policy aggregation in optimization under uncertainty. Math. Oper. Res. 16(1), 119–147 (1991)

    MATH  Google Scholar 

  18. Royden, H.L.: Real Analysis. Macmillan Publishing Co., London, 3rd edition, 1988

  19. Sahinidis, N.V.: BARON: A general purpose global optimization software package. J. Global Optim. 8, 201–205 (1996)

    MATH  Google Scholar 

  20. Schultz, R.: On structure and stability in stochastic programs with random technology matrix and complete integer recourse. Math. Program. 70(1), 73–89 (1995)

    Article  Google Scholar 

  21. Schultz, R., Stougie, L., van der Vlerk, M.H.: Solving stochastic programs with integer recourse by enumeration: A framework using Gröbner basis reductions. Math. Program. 83, 229–252 (1998)

    Article  MathSciNet  Google Scholar 

  22. Tawarmalani, M., Sahinidis, N.V.: Convexification and Global Optimization in Continuous and Mixed-Integer Nonlinear Programming: Theory, Algorithms, Software, and Applications. Kluwer Academic Publishers, Nonconvex Optimization and Its Applications Series, Vol 65, Dordrecht, 2002

  23. Van Slyke, R., Wets, R.J.-B.: L-Shaped linear programs with applications to optimal control and stochastic programming. SIAM J. Appl. Math. 17, 638–663 (1969)

    MATH  Google Scholar 

  24. Wets, R.J-B.: Programming under uncertainty: The solution set. SIAM J. Appl. Math. 14, 1143–1151 (1966)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Industrial & Systems Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

    Shabbir Ahmed

  2. Krannert School of Management, Purdue University, West Lafayette, IN 47907, USA

    Mohit Tawarmalani

  3. Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

    Nikolaos V. Sahinidis

Authors
  1. Shabbir Ahmed
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohit Tawarmalani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nikolaos V. Sahinidis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shabbir Ahmed.

Additional information

The authors wish to acknowledge partial financial support from the IBM Research Division, ExxonMobil Upstream Research Company, and the National Science Foundation under awards DMI 95-02722, DMI 00-99726, and DMI 01-15166

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ahmed, S., Tawarmalani, M. & Sahinidis, N. A finite branch-and-bound algorithm for two-stage stochastic integer programs. Math. Program., Ser. A 100, 355–377 (2004). https://doi.org/10.1007/s10107-003-0475-6

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-003-0475-6

Keywords

Search

Navigation