Skip to main content
SpringerLink
Log in

A class of algorithms for mixed-integer bilevel min–max optimization

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

Abstract

In this paper, we introduce a new class of algorithms for solving the mixed-integer bilevel min–max optimization problem. This problem involves two players, a leader and a follower, who play a Stackelberg game. In particular, the leader seeks to minimize over a set of discrete variables the maximum objective that the follower can achieve. The complicating features of our problem are that a subset of the follower’s decisions are restricted to be integer-valued, and that the follower’s decisions are constrained by the leader’s decisions. We first describe several bilevel min–max programs that can be used to obtain lower and upper bounds on the optimal objective value of the problem. We then present algorithms for this problem that finitely terminate with an optimal solution when the leader variables are restricted to take binary values. Finally, we report the results of a computational study aimed at evaluating the quality of our algorithms on two families of randomly generated problems.

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
Fig. 2

Similar content being viewed by others

References

  1. Ball, M.O., Golden, B.L., Vohra, R.V.: Finding the most vital arcs in a network. Oper. Res. Lett. 8(2), 73–76 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bard, J.F., Moore, J.T.: An algorithm for the discrete bilevel programming problem. Nav. Res. Logist. 39(3), 419–435 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bazaraa, M.S., Jarvis, J.J., Sherali, H.D.: Linear Programming and Network Flows, 3rd edn. Wiley, Hoboken (2005)

    MATH  Google Scholar 

  4. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, Hoboken (2006)

    Book  MATH  Google Scholar 

  5. Beheshti, B., Özaltın, O.Y., Zare, M.H., Prokopyev, O.A.: Exact solution approach for a class of nonlinear bilevel knapsack problems. J. Glob. Optim. (2014). doi:10.1007/s10898-014-0189-8

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

    Article  MATH  Google Scholar 

  7. Bialas, W.F., Karwan, M.H.: Two-level linear programming. Manag. Sci. 30(8), 1004–1021 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  8. Brotcorne, L., Hanafi, S., Mansi, R.: A Dynamic programming algorithm for the Bilevel Knapsack Problem. Oper. Res. Lett. 37(3), 215–218 (2009). ISSN 0167–6377

    Article  MathSciNet  MATH  Google Scholar 

  9. Brotcorne, L., Hanafi, S., Mansi, R.: One-level reformulation of the bilevel knapsack problem using dynamic programming. Discrete Optim. 10(1), 1–10 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brown, G.G., Carlyle, W.M., Salmerón, J., Wood, K.: Analyzing the vulnerability of critical infrastructure to attack and planning defenses. In: Greenberg, H.J., Smith, J.C. (eds.) Tutorials in Operations Research: Emerging Theory, Methods, and Applications, pp. 102–123. INFORMS, Hanover, MD (2005)

    Google Scholar 

  11. Candler, W., Townsley, R.: A linear two-level programming problem. Comput. Oper. Res. 9(1), 59–76 (1982)

    Article  MathSciNet  Google Scholar 

  12. Caprara, A., Carvalho, M., Lodi, A., Woeginger, G.J.: A complexity and approximability study of the bilevel knapsack problem. In: Goemans, M., Correa, J. (eds.) Integer Programming and Combinatorial Optimization, pp. 98–109. Springer, Berlin (2013)

    Google Scholar 

  13. Corley, H.W., Sha, D.Y.: Most vital links and nodes in weighted networks. Oper. Res. Lett. 1(4), 157–160 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  14. Danskin, J.: The theory of max–min, with applications. SIAM J. Appl. Math. 14, 641–664 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  15. Delgadillo, A., Arroyo, J.M., Alguacil, N.: Analysis of electric grid interdiction with line switching. IEEE Trans. Power Syst. 25(2), 633–641 (2010)

    Article  Google Scholar 

  16. Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic Publishers, Boston (2002)

    MATH  Google Scholar 

  17. Dempe, S., Richter, K.: Bilevel programming with knapsack constraints. CEJOR 8(2), 93–107 (2000)

    MathSciNet  MATH  Google Scholar 

  18. DeNegre, S.T., Ralphs, T.K.: A branch-and-cut algorithm for integer bilevel linear programs. In: Chinneck, J.W., Kristjansson, B., Saltzman, M.J. (eds.) Operations Research and Cyber-Infrastructure, pp. 65–78. Springer, New York (2009)

    Chapter  Google Scholar 

  19. Fulkerson, D.R., Harding, G.C.: Maximizing minimum source-sink path subject to a budget constraint. Math. Program. 13(1), 116–118 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  20. Glover, F.: Improved linear integer programming formulations of nonlinear integer problems. Manag. Sci. 22(4), 455–469 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  21. Glover, F., Woolsey, E.: Converting the 0–1 polynomial programming problem to a 0–1 linear program. Oper. Res. 22(1), 180–182 (1974)

    Article  MATH  Google Scholar 

  22. Golden, B.: A problem in network interdiction. Nav. Res. Logist. Q. 25(4), 711–713 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hansen, P., Jaumard, B., Savard, G.: New branch-and-bound rules for linear bilevel programming. SIAM J. Sci. Stat. Comput. 13(5), 1194–1217 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Israeli, E., Wood, R.K.: Shortest-path network interdiction. Networks 40(2), 97–111 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lim, C., Smith, J.C.: Algorithms for discrete and continuous multicommodity flow network interdiction problems. IIE Trans. 39(1), 15–26 (2007)

    Article  Google Scholar 

  26. Mahdavi Pajouh, F., Boginski, V., Pasiliao, E.L.: Minimum vertex blocker clique problem. Networks 64(1), 48–64 (2014)

    Article  MathSciNet  Google Scholar 

  27. Malaviya, A., Rainwater, C., Sharkey, T.C.: Multi-period network interdiction problems with applications to city-level drug enforcement. IIE Trans. 44(5), 368–380 (2012)

    Article  Google Scholar 

  28. Malik, K., Mittal, A.K., Gupta, S.K.: The k most vital arcs in the shortest path problem. Oper. Res. Lett. 8(4), 223–227 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  29. MIT Strategic Engineering Research Group. MATLAB tools for network analysis. http://strategic.mit.edu/docs/matlab_networks/random_graph.m. Accessed Oct 2013

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Morton, D.P., Pan, F., Saeger, K.J.: Models for nuclear smuggling interdiction. IIE Trans. 39(1), 3–14 (2007)

    Article  Google Scholar 

  32. Ratliff, H.D., Sicilia, G.T., Lubore, S.H.: Finding the n most vital links in flow networks. Manag. Sci. 21(5), 531–539 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  33. Vazirani, V.V.: Approximation Algorithms. Springer, New York (2004)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. von Stackelberg, H.: The Theory of the Market Economy. William Hodge and Co., London (1952)

    Google Scholar 

  36. Ward, J.E., Wendell, R.E.: Approaches to sensitivity analysis in linear programming. Ann. Oper. Res. 27, 3–38 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  37. Wets, R.J.-B.: On the continuity of the value of a linear program and of related polyhedral-value multifunctions. Math. Program. Study 24, 14–29 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  38. Wollmer, R.D.: Removing arcs from a network. Oper. Res. 12(6), 934–940 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  39. Wood, R.K.: Deterministic network interdiction. Math. Comput. Model. 17(2), 1–18 (1993)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL, 32611, USA

    Yen Tang, Jean-Philippe P. Richard & J. Cole Smith

Authors
  1. Yen Tang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jean-Philippe P. Richard
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Cole Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jean-Philippe P. Richard.

Additional information

The authors gratefully acknowledge an anonymous referee, whose remarks led to an improved presentation of our research. This work has been supported by the National Science Foundation through Grant CMMI-11000765, the Defense Threat Reduction Agency through Grant HDTRA1-10-1-0050, the Air Force Office of Scientific Research under Grant FA9550-12-1-0353, and the Office of Naval Research under grant N000141310036.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tang, Y., Richard, JP.P. & Smith, J.C. A class of algorithms for mixed-integer bilevel min–max optimization. J Glob Optim 66, 225–262 (2016). https://doi.org/10.1007/s10898-015-0274-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10898-015-0274-7

Keywords

Search

Navigation