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New concepts and an algorithm for multiobjective bilevel programming: optimistic, pessimistic and moderate solutions

Author

Listed:
  • Maria João Alves

    (University of Coimbra
    INESC Coimbra)

  • Carlos Henggeler Antunes

    (University of Coimbra, Polo 2
    INESC Coimbra)

  • João Paulo Costa

    (University of Coimbra
    INESC Coimbra)

Abstract

Bilevel optimization deals with hierarchical mathematical programming problems in which two decision makers, the leader and the follower, control different sets of variables and have their own objective functions subject to interdependent constraints. Whenever multiple objective functions exist at the lower-level problem, the leader should cope with the uncertainty pertaining to the follower’s reaction. The leader can adopt a more optimistic or more pessimistic stance regarding the follower’s choice within his efficient region, which is restricted by the leader’s choice. Moreover, the leader may also have multiple objective functions. This paper presents new concepts associated with solutions to problems with multiple objective functions at the lower-level and a single or multiple objective functions at the upper-level, exploring the optimistic and pessimistic leader’s perspectives and their interplay with the follower’s choices. Extreme solutions (called optimistic/deceiving and pessimistic/rewarding) and a moderate solution, resulting from the risk the leader is willing to accept, are defined for problems with a single objective at the upper-level (semivectorial problems). Definitions of optimistic and pessimistic Pareto fronts are proposed for problems with multiple objective functions at the upper-level. These novel concepts are illustrated emphasizing the difficulties associated with the computation of those solutions. In addition, a differential evolution algorithm, approximating the extreme and moderate solutions for the semivectorial problem, is presented. Illustrative results of this algorithm further stress the challenges and pitfalls associated with the computation and interpretation of results in this kind of problems, which have not been properly addressed in literature and may lead to misleading conclusions.

Suggested Citation

  • Maria João Alves & Carlos Henggeler Antunes & João Paulo Costa, 2021. "New concepts and an algorithm for multiobjective bilevel programming: optimistic, pessimistic and moderate solutions," Operational Research, Springer, vol. 21(4), pages 2593-2626, December.
    • Handle: RePEc:spr:operea:v:21:y:2021:i:4:d:10.1007_s12351-019-00534-9
    DOI: 10.1007/s12351-019-00534-9
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    References listed on IDEAS

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    1. Benoît Colson & Patrice Marcotte & Gilles Savard, 2007. "An overview of bilevel optimization," Annals of Operations Research, Springer, vol. 153(1), pages 235-256, September.
    2. Calvete, Herminia I. & Galé, Carmen, 2011. "On linear bilevel problems with multiple objectives at the lower level," Omega, Elsevier, vol. 39(1), pages 33-40, January.
    3. Ankhili, Z. & Mansouri, A., 2009. "An exact penalty on bilevel programs with linear vector optimization lower level," European Journal of Operational Research, Elsevier, vol. 197(1), pages 36-41, August.
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    Cited by:

    1. 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.
    2. Alves, Maria João & Henggeler Antunes, Carlos, 2022. "A new exact method for linear bilevel problems with multiple objective functions at the lower level," European Journal of Operational Research, Elsevier, vol. 303(1), pages 312-327.

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