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Bilevel Integer Programs with Stochastic Right-Hand Sides

Author

Listed:
  • Junlong Zhang

    (Department of Industrial Engineering, Tsinghua University, Beijing 100084, China)

  • Osman Y. Özaltın

    (Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, North Carolina 27695)

Abstract

We develop an exact value function-based approach to solve a class of bilevel integer programs with stochastic right-hand sides. We first study structural properties and design two methods to efficiently construct the value function of a bilevel integer program. Most notably, we generalize the integer complementary slackness theorem to bilevel integer programs. We also show that the value function of a bilevel integer program can be characterized by its values on a set of so-called bilevel minimal vectors. We then solve the value function reformulation of the original bilevel integer program with stochastic right-hand sides using a branch-and-bound algorithm. We demonstrate the performance of our solution methods on a set of randomly generated instances. We also apply the proposed approach to a bilevel facility interdiction problem. Our computational experiments show that the proposed solution methods can efficiently optimize large-scale instances. The performance of our value function-based approach is relatively insensitive to the number of scenarios, but it is sensitive to the number of constraints with stochastic right-hand sides. Summary of Contribution: Bilevel integer programs arise in many different application areas of operations research including supply chain, energy, defense, and revenue management. This paper derives structural properties of the value functions of bilevel integer programs. Furthermore, it proposes exact solution algorithms for a class of bilevel integer programs with stochastic right-hand sides. These algorithms extend the applicability of bilevel integer programs to a larger set of decision-making problems under uncertainty.

Suggested Citation

  • Junlong Zhang & Osman Y. Özaltın, 2021. "Bilevel Integer Programs with Stochastic Right-Hand Sides," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1644-1660, October.
    • Handle: RePEc:inm:orijoc:v:33:y:2021:i:4:p:1644-1660
    DOI: 10.1287/ijoc.2020.1055
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    References listed on IDEAS

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    1. Holmberg, Kaj & Ronnqvist, Mikael & Yuan, Di, 1999. "An exact algorithm for the capacitated facility location problems with single sourcing," European Journal of Operational Research, Elsevier, vol. 113(3), pages 544-559, March.
    2. S. Dempe & F. Mefo Kue, 2017. "Solving discrete linear bilevel optimization problems using the optimal value reformulation," Journal of Global Optimization, Springer, vol. 68(2), pages 255-277, June.
    3. Matteo Fischetti & Ivana Ljubić & Michele Monaci & Markus Sinnl, 2017. "A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs," Operations Research, INFORMS, vol. 65(6), pages 1615-1637, December.
    4. Robert G. Jeroslow, 1978. "Some Basis Theorems for Integral Monoids," Mathematics of Operations Research, INFORMS, vol. 3(2), pages 145-154, May.
    5. Bard, Jonathan F. & Plummer, John & Claude Sourie, Jean, 2000. "A bilevel programming approach to determining tax credits for biofuel production," European Journal of Operational Research, Elsevier, vol. 120(1), pages 30-46, January.
    6. Benoît Colson & Patrice Marcotte & Gilles Savard, 2007. "An overview of bilevel optimization," Annals of Operations Research, Springer, vol. 153(1), pages 235-256, September.
    7. Leonardo Lozano & J. Cole Smith, 2017. "A Value-Function-Based Exact Approach for the Bilevel Mixed-Integer Programming Problem," Operations Research, INFORMS, vol. 65(3), pages 768-786, June.
    8. Gerald Brown & Matthew Carlyle & Javier Salmerón & Kevin Wood, 2006. "Defending Critical Infrastructure," Interfaces, INFORMS, vol. 36(6), pages 530-544, December.
    9. Onur Tavaslıoğlu & Oleg A. Prokopyev & Andrew J. Schaefer, 2019. "Solving Stochastic and Bilevel Mixed-Integer Programs via a Generalized Value Function," Operations Research, INFORMS, vol. 67(6), pages 1659-1677, November.
    10. Wayne F. Bialas & Mark H. Karwan, 1984. "Two-Level Linear Programming," Management Science, INFORMS, vol. 30(8), pages 1004-1020, August.
    11. C. Audet & P. Hansen & B. Jaumard & G. Savard, 1997. "Links Between Linear Bilevel and Mixed 0–1 Programming Problems," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 273-300, May.
    12. Jonathan F. Bard & James T. Moore, 1992. "An algorithm for the discrete bilevel programming problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 39(3), pages 419-435, April.
    13. Bo Zeng, 2020. "A Practical Scheme to Compute the Pessimistic Bilevel Optimization Problem," INFORMS Journal on Computing, INFORMS, vol. 32(4), pages 1128-1142, October.
    14. Chiou, Suh-Wen, 2005. "Bilevel programming for the continuous transport network design problem," Transportation Research Part B: Methodological, Elsevier, vol. 39(4), pages 361-383, May.
    15. Afshin Kamyabniya & M. M. Lotfi & Hua Cai & Hasan Hosseininasab & Saeed Yaghoubi & Yuehwern Yih, 2019. "A two-phase coordinated logistics planning approach to platelets provision in humanitarian relief operations," IISE Transactions, Taylor & Francis Journals, vol. 51(1), pages 1-21, January.
    16. Dajun Yue & Jiyao Gao & Bo Zeng & Fengqi You, 2019. "A projection-based reformulation and decomposition algorithm for global optimization of a class of mixed integer bilevel linear programs," Journal of Global Optimization, Springer, vol. 73(1), pages 27-57, January.
    17. James T. Moore & Jonathan F. Bard, 1990. "The Mixed Integer Linear Bilevel Programming Problem," Operations Research, INFORMS, vol. 38(5), pages 911-921, October.
    18. Alizadeh, S.M. & Marcotte, P. & Savard, G., 2013. "Two-stage stochastic bilevel programming over a transportation network," Transportation Research Part B: Methodological, Elsevier, vol. 58(C), pages 92-105.
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