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The New Solution Concept to Ill-Posed Bilevel Programming: Non-Antagonistic Pessimistic Solution

Author

Listed:
  • Xiang Li

    (State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China)

  • Tiesong Hu

    (State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China)

  • Xin Wang

    (Hubei Provincial Water Saving Research Center, Hubei Water Resources Research Institute, Wuhan 430070, China)

  • Ali Mahmoud

    (State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan 430072, China)

  • Xiang Zeng

    (School of Resource and Environmental Sciences, Wuhan University, Wuhan 430079, China)

Abstract

It is hardly realistic to assume that, under all decision circumstances, followers will always choose a solution that leads to the worst upper-level objective functional value. However, this generally accepted concept of the pessimistic solution to the ill-posed bilevel programming problems may lead to the leader’s attitude being more pessimistic vis à vis his anticipation of the follower’s decision being non-antagonistic. It will result in a wrong pessimistic solution and a greater potential of cooperation space between the leader and the followers. This paper presents a new concept of a non-antagonistic pessimistic solution with four numerical examples for bilevel programming problems from a non-antagonistic point of view. We prove that the objective function value of the non-antagonistic pessimistic solution generally dominates or is equal to the objective functional value of the pessimistic solution and the rewarding solution, and the maximum potential space for leader-follower cooperation can be overestimated in a generally applied pessimistic solution. Our research extends the concept of the pessimistic solution. It also sheds light on the insights that the non-antagonistic pessimistic solution can describe the practical potential of cooperation space between the leader and followers in non-antagonistic circumstances.

Suggested Citation

  • Xiang Li & Tiesong Hu & Xin Wang & Ali Mahmoud & Xiang Zeng, 2023. "The New Solution Concept to Ill-Posed Bilevel Programming: Non-Antagonistic Pessimistic Solution," Mathematics, MDPI, vol. 11(6), pages 1-13, March.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:6:p:1422-:d:1098096
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    References listed on IDEAS

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