IDEAS home Printed from https://ideas.repec.org/a/inm/ordeca/v17y2020i1p74-95.html
   My bibliography  Save this article

On Bilevel Optimization with Inexact Follower

Author

Listed:
  • M. Hosein Zare

    (Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15260)

  • Oleg A. Prokopyev

    (Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, Pennsylvania 15260)

  • Denis Sauré

    (Department of Industrial Engineering, University of Chile, 8370456 Santiago, Chile)

Abstract

Traditionally, in the bilevel optimization framework, a leader chooses her actions by solving an upper-level problem, assuming that a follower chooses an optimal reaction by solving a lower-level problem. However, in many settings, the lower-level problems might be nontrivial, thus requiring the use of tailored algorithms for their solution. More importantly, in practice, such problems might be inexactly solved by heuristics and approximation algorithms. Motivated by this consideration, we study a broad class of bilevel optimization problems where the follower might not optimally react to the leader’s actions. In particular, we present a modeling framework in which the leader considers that the follower might use one of a number of known algorithms to solve the lower-level problem, either approximately or heuristically. Thus, the leader can hedge against the follower’s use of suboptimal solutions. We provide algorithmic implementations of the framework for a class of nonlinear bilevel knapsack problem (BKP), and we illustrate the potential impact of incorporating this realistic feature through numerical experiments in the context of defender-attacker problems.

Suggested Citation

  • M. Hosein Zare & Oleg A. Prokopyev & Denis Sauré, 2020. "On Bilevel Optimization with Inexact Follower," Decision Analysis, INFORMS, vol. 17(1), pages 74-95, March.
    • Handle: RePEc:inm:ordeca:v:17:y:2020:i:1:p:74-95
    DOI: 10.1287/deca.2019.0392
    as

    Download full text from publisher

    File URL: https://doi.org/10.1287/deca.2019.0392
    Download Restriction: no
    File URL: https://libkey.io/10.1287/deca.2019.0392?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. Dimitris Bertsimas & Melvyn Sim, 2004. "The Price of Robustness," Operations Research, INFORMS, vol. 52(1), pages 35-53, February.
    2. 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.
    3. Mohammad E. Nikoofal & Jun Zhuang, 2012. "Robust Allocation of a Defensive Budget Considering an Attacker's Private Information," Risk Analysis, John Wiley & Sons, vol. 32(5), pages 930-943, May.
    4. Juan S. Borrero & Oleg A. Prokopyev & Denis Sauré, 2016. "Sequential Shortest Path Interdiction with Incomplete Information," Decision Analysis, INFORMS, vol. 13(1), pages 68-98, March.
    5. Egon Balas & Eitan Zemel, 1980. "An Algorithm for Large Zero-One Knapsack Problems," Operations Research, INFORMS, vol. 28(5), pages 1130-1154, October.
    6. Silvano Martello & Paolo Toth, 1988. "A New Algorithm for the 0-1 Knapsack Problem," Management Science, INFORMS, vol. 34(5), pages 633-644, May.
    7. Juan S. Borrero & Oleg A. Prokopyev & Denis Sauré, 2019. "Sequential Interdiction with Incomplete Information and Learning," Operations Research, INFORMS, vol. 67(1), pages 72-89, January.
    8. 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.
    9. Robin M. Hogarth & Natalia Karelaia, 2005. "Simple Models for Multiattribute Choice with Many Alternatives: When It Does and Does Not Pay to Face Trade-offs with Binary Attributes," Management Science, INFORMS, vol. 51(12), pages 1860-1872, December.
    10. Benoît Colson & Patrice Marcotte & Gilles Savard, 2007. "An overview of bilevel optimization," Annals of Operations Research, Springer, vol. 153(1), pages 235-256, September.
    11. M. Hosein Zare & Juan S. Borrero & Bo Zeng & Oleg A. Prokopyev, 2019. "A note on linearized reformulations for a class of bilevel linear integer problems," Annals of Operations Research, Springer, vol. 272(1), pages 99-117, January.
    12. Cao, D. & Leung, L. C., 2002. "A partial cooperation model for non-unique linear two-level decision problems," European Journal of Operational Research, Elsevier, vol. 140(1), pages 134-141, July.
    13. Nikoofal, Mohammad E. & Zhuang, Jun, 2015. "On the value of exposure and secrecy of defense system: First-mover advantage vs. robustness," European Journal of Operational Research, Elsevier, vol. 246(1), pages 320-330.
    14. Alberto Caprara & Margarida Carvalho & Andrea Lodi & Gerhard J. Woeginger, 2016. "Bilevel Knapsack with Interdiction Constraints," INFORMS Journal on Computing, INFORMS, vol. 28(2), pages 319-333, May.
    15. Gerald Brown & Matthew Carlyle & Javier Salmerón & Kevin Wood, 2006. "Defending Critical Infrastructure," Interfaces, INFORMS, vol. 36(6), pages 530-544, December.
    16. Martello, Silvano & Toth, Paolo, 1977. "An upper bound for the zero-one knapsack problem and a branch and bound algorithm," European Journal of Operational Research, Elsevier, vol. 1(3), pages 169-175, May.
    17. Andrew Samuel & Seth D. Guikema, 2012. "Resource Allocation for Homeland Defense: Dealing with the Team Effect," Decision Analysis, INFORMS, vol. 9(3), pages 238-252, September.
    18. Zhang, Jing & Zhuang, Jun & Behlendorf, Brandon, 2018. "Stochastic shortest path network interdiction with a case study of Arizona–Mexico border," Reliability Engineering and System Safety, Elsevier, vol. 179(C), pages 62-73.
    19. Shan, Xiaojun & Zhuang, Jun, 2013. "Hybrid defensive resource allocations in the face of partially strategic attackers in a sequential defender–attacker game," European Journal of Operational Research, Elsevier, vol. 228(1), pages 262-272.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Hughes, Michael S. & Lunday, Brian J., 2022. "The Weapon Target Assignment Problem: Rational Inference of Adversary Target Utility Valuations from Observed Solutions," Omega, Elsevier, vol. 107(C).
    2. Beck, Yasmine & Ljubić, Ivana & Schmidt, Martin, 2023. "A survey on bilevel optimization under uncertainty," European Journal of Operational Research, Elsevier, vol. 311(2), pages 401-426.
    3. Pferschy, Ulrich & Nicosia, Gaia & Pacifici, Andrea & Schauer, Joachim, 2021. "On the Stackelberg knapsack game," European Journal of Operational Research, Elsevier, vol. 291(1), pages 18-31.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Beck, Yasmine & Ljubić, Ivana & Schmidt, Martin, 2023. "A survey on bilevel optimization under uncertainty," European Journal of Operational Research, Elsevier, vol. 311(2), pages 401-426.
    2. Juan S. Borrero & Oleg A. Prokopyev & Denis Sauré, 2019. "Sequential Interdiction with Incomplete Information and Learning," Operations Research, INFORMS, vol. 67(1), pages 72-89, January.
    3. M. Hosein Zare & Osman Y. Özaltın & Oleg A. Prokopyev, 2018. "On a class of bilevel linear mixed-integer programs in adversarial settings," Journal of Global Optimization, Springer, vol. 71(1), pages 91-113, May.
    4. Hunt, Kyle & Zhuang, Jun, 2024. "A review of attacker-defender games: Current state and paths forward," European Journal of Operational Research, Elsevier, vol. 313(2), pages 401-417.
    5. Qingqing Zhai & Rui Peng & Jun Zhuang, 2020. "Defender–Attacker Games with Asymmetric Player Utilities," Risk Analysis, John Wiley & Sons, vol. 40(2), pages 408-420, February.
    6. 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.
    7. Jing Yang & Juan S. Borrero & Oleg A. Prokopyev & Denis Sauré, 2021. "Sequential Shortest Path Interdiction with Incomplete Information and Limited Feedback," Decision Analysis, INFORMS, vol. 18(3), pages 218-244, September.
    8. Juan S. Borrero & Leonardo Lozano, 2021. "Modeling Defender-Attacker Problems as Robust Linear Programs with Mixed-Integer Uncertainty Sets," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1570-1589, October.
    9. 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.
    10. Zhang, Jing & Zhuang, Jun, 2019. "Modeling a multi-target attacker-defender game with multiple attack types," Reliability Engineering and System Safety, Elsevier, vol. 185(C), pages 465-475.
    11. David Pisinger, 1999. "Core Problems in Knapsack Algorithms," Operations Research, INFORMS, vol. 47(4), pages 570-575, August.
    12. Hunt, Kyle & Agarwal, Puneet & Zhuang, Jun, 2022. "On the adoption of new technology to enhance counterterrorism measures: An attacker–defender game with risk preferences," Reliability Engineering and System Safety, Elsevier, vol. 218(PB).
    13. Karwowski, Jan & Mańdziuk, Jacek, 2019. "A Monte Carlo Tree Search approach to finding efficient patrolling schemes on graphs," European Journal of Operational Research, Elsevier, vol. 277(1), pages 255-268.
    14. Wei, Ningji & Walteros, Jose L., 2022. "Integer programming methods for solving binary interdiction games," European Journal of Operational Research, Elsevier, vol. 302(2), pages 456-469.
    15. M. Hosein Zare & Juan S. Borrero & Bo Zeng & Oleg A. Prokopyev, 2019. "A note on linearized reformulations for a class of bilevel linear integer problems," Annals of Operations Research, Springer, vol. 272(1), pages 99-117, January.
    16. Nageswara S. V. Rao & Chris Y. T. Ma & Fei He & David K. Y. Yau & Jun Zhuang, 2018. "Cyber–Physical Correlation Effects in Defense Games for Large Discrete Infrastructures," Games, MDPI, vol. 9(3), pages 1-24, July.
    17. Keskin, Burcu B. & Griffin, Emily C. & Prell, Jonathan O. & Dilkina, Bistra & Ferber, Aaron & MacDonald, John & Hilend, Rowan & Griffis, Stanley & Gore, Meredith L., 2023. "Quantitative Investigation of Wildlife Trafficking Supply Chains: A Review," Omega, Elsevier, vol. 115(C).
    18. Yan, Xihong & Ren, Xiaorong & Nie, Xiaofeng, 2022. "A budget allocation model for domestic airport network protection," Socio-Economic Planning Sciences, Elsevier, vol. 82(PB).
    19. Ketkov, Sergey S. & Prokopyev, Oleg A., 2020. "On greedy and strategic evaders in sequential interdiction settings with incomplete information," Omega, Elsevier, vol. 92(C).
    20. Vladimir Stozhkov & Vladimir Boginski & Oleg A. Prokopyev & Eduardo L. Pasiliao, 2017. "A simple greedy heuristic for linear assignment interdiction," Annals of Operations Research, Springer, vol. 249(1), pages 39-53, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:ordeca:v:17:y:2020:i:1:p:74-95. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.