Skip to main content
Springer Nature Link
Log in

Enhanced exact algorithms for discrete bilevel linear problems

  • Original Paper
  • Published:
Optimization Letters Aims and scope Submit manuscript

Abstract

We address a particular class of bilevel linear programming problems in which all the variables are discrete. The main computational complexities are analyzed and two enhanced exact algorithms are proposed. The rationale behind these two algorithms is described and a modified version is presented for both. A common test bed is used to assess their computational efficiency along with a comparison with an existing benchmark algorithm.

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
Fig. 5

Similar content being viewed by others

References

  1. Audet, C., Haddad, J., Savard, G.: Disjunctive cuts for continuous linear bilevel programming. Optim. Lett. 1(3), 259–267 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  2. Audet, C., Hansen, P., Jaumard, B., Savard, G.: Links between linear bilevel and mixed 0–1 programming problems. J. Optim. Theory. Appl. 93(2), 273–300 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  3. Audet, C., Savard, G., Zghal, W.: New branch-and-cut algorithm for bilevel linear programming. J. Optim. Theory. Appl. 134(2), 353–370 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bard, J.F.: Practical bilevel optimization: algorithms and applications. Kluwer Academic Publishers, Dordrecht (1998)

    Book  MATH  Google Scholar 

  5. Bard, J.F., Moore, J.T.: A branch and bound algorithm for the bilevel programming problem. SIAM. J. Sci. Stat. Comput. 11(2), 281–292 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  6. Bialas, W., Karwan, M.: On two-level optimization. IEEE. Trans. Autom. Control. 27(1), 211–214 (1982)

    Article  MATH  Google Scholar 

  7. Bialas, W., Karwan, M.: Two-level linear programming. Manag. Sci. 30(8), 1004–1020 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bianco, L., Caramia, M., Giordani, S.: A bilevel flow model for hazmat transportation network design. Transp. Res. Part. C. 17, 175–196 (2009)

    Article  Google Scholar 

  9. Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Annals. Oper. Res. 153(1), 235–256 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dempe, S.: Foundation of bilevel programming. Kluwer Academic Publications (2002)

  11. DeNegre, S.T., Ralphs, T.K.: A branch-and-cut algorithm for integer bilevel linear programs. Oper. Res. Cyber. Infrastruct. 47(2), 65–78 (2009)

    Article  Google Scholar 

  12. Fortuny-Amat, J., McCarl, B.: A representation and economic interpretation of a two-level programming problem. J. Oper. Res. Soc. 32(9), 783–792 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jeroslow, R.G.: The polynomial hierarchy and a simple model for competitive analysis. Math. Program. 32(2), 146–164 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  14. Koch, T., Achterberg, T., Andersen, E., Bastert, O., Berthold, T., Bixby, R.E., Danna, E., Gamrath, G., Gleixner, A.M., Heinz, S., Lodi, A., Mittelmann, H., Ralphs, T., Salvagnin, D., Steffy, D.E., Wolter, K.: MIPLIB 2010—mixed integer programming library version 5. Math. Program. Comput. 3(2), 103–163 (2011)

    Article  MathSciNet  Google Scholar 

  15. Mansi, R., Alves, C., de Carvalho, J.M.V., Hanafi, S.: An exact algorithm for bilevel 0–1 knapsack problems. Hindawi Publishing Corporation Mathematical Problems in Engineering, (2012)

  16. Moore, J.T., Bard, J.F.: The mixed integer linear bilevel programming problem. Oper. Res. 38(5), 911–921 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  17. Shiquan, W.U., Yang, C., Marcotte, P.: A cutting plane method for linear bilevel programs. Syst. Sci. Math. Sci. 11(2), 125–133 (1998)

    MATH  Google Scholar 

  18. Vicente, L.N., Savard, G., Judice, J.J.: Discrete linear bilevel programming problem. J. Optim. Theory. Appl. 89(3), 597–614 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  19. Wen, U.P., Hsu, S.T.: Linear bi-level programming problems—a review. J. Oper. Res. Soc. 42(2), 125–133 (1991)

    MATH  Google Scholar 

  20. Wen, U.P., Huang, A.D.: A simple tabu search method to solve the mixed-integer linear bilevel programming problem. Eur. J. Oper. Res. 88(3), 563–571 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Wen, U.P., Yang, Y.H.: Algorithms for solving the mixed integer two-level linear programming problem. Comput. Oper. Res. 17(2), 133–142 (1990)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria dell’Impresa, University of Rome “Tor Vergata”, Via del Politecnico, 1, 00133, Rome, Italy

    Massimiliano Caramia & Renato Mari

Authors
  1. Massimiliano Caramia
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Renato Mari
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Massimiliano Caramia.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Caramia, M., Mari, R. Enhanced exact algorithms for discrete bilevel linear problems. Optim Lett 9, 1447–1468 (2015). https://doi.org/10.1007/s11590-015-0872-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-015-0872-9

Keywords