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A Study on Optimistic and Pessimistic Pareto-Fronts in Multiobjective Bilevel Optimization via \(\delta \)-Perturbation

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Evolutionary Multi-Criterion Optimization (EMO 2025)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15512))

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Abstract

In bilevel optimization, an upper level (UL) decision maker seeks to optimize an objective function while considering the optimal solutions of a lower level (LL) optimization problem. This hierarchical structure poses modeling and solution challenges, especially when the LL problem has multiple solutions. In such a case, the UL needs to make assumptions about the LL reaction. In the optimistic approach, the UL assumes that the LL reaction will be favorable, while in the pessimistic approach the opposite is true. In this study, we consider the case of a multiobjective bilevel optimization problem, where the UL has multiple objectives, while the LL has a single objective, but multiple optimal solutions for any given UL decision. Given that the LL can choose any solution from its optimal set, and in case the UL is not aware of the LL choice function, it leads to the possibility of two Pareto-optimal fronts at the UL, i.e. the optimistic and the pessimistic frontiers. To approximate both Pareto-optimal fronts, a \(\delta \)-perturbation approximation is proposed in this paper, where the LL objective is perturbed by a small \(\delta \) by utilizing the UL objectives at the LL. The perturbed reformulated bilevel problem is then solved via an extension of m-BLEAQ, an evolutionary bilevel algorithm that can deal with bilevel problems with multiple objectives at the UL and a single objective at the LL. The application of the m-BLEAQ algorithm to the reformulated bilevel problem leads to the identification of the optimistic and pessimistic frontiers for the multiobjective bilevel problem. In this proof-of-concept study, the proposed reformulation strategy is demonstrated on two test problems, showing the effectiveness of the proposal in accurately finding the optimistic and pessimistic frontiers.

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Notes

  1. 1.

    In semivectorial bilevel optimization, the UL problem is single objective, while the LL problem is multiobjective.

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Acknowledgements

The authors acknowledge the financial support from the Slovenian Research Agency (research core funding No. P2-0098). This work was also partially funded by the European Union’s Horizon Europe research an innovation programme under Grant Agreement No 101077049 (CONDUCTOR).

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Authors and Affiliations

  1. Computer Systems Department, Jožef Stefan Institute, Jamova c. 39, Ljubljana, Slovenia

    Margarita Antoniou & Gregor Papa

  2. Operations and Decision Sciences, Indian Institute of Management Ahmedabad, Gujarat, India

    Ankur Sinha

  3. Jožef Stefan International Postgraduate School, Jamova c. 39, Ljubljana, Slovenia

    Margarita Antoniou & Gregor Papa

Authors
  1. Margarita Antoniou
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  2. Ankur Sinha
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  3. Gregor Papa
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Correspondence to Margarita Antoniou .

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Editors and Affiliations

  1. University of New South Wales, Canberra, NSW, Australia

    Hemant Singh

  2. University of New South Wales, Canberra, NSW, Australia

    Tapabrata Ray

  3. SLB Cambridge Research, Cambridge, UK

    Joshua Knowles

  4. RMIT, Melbourne, VIC, Australia

    Xiaodong Li

  5. University of Warwick, Coventry, Warwickshire, UK

    Juergen Branke

  6. University of New South Wales, Canberra, NSW, Australia

    Bing Wang

  7. Japan Aerospace Exploration Agency, Tokyo, Japan

    Akira Oyama

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Antoniou, M., Sinha, A., Papa, G. (2025). A Study on Optimistic and Pessimistic Pareto-Fronts in Multiobjective Bilevel Optimization via \(\delta \)-Perturbation. In: Singh, H., et al. Evolutionary Multi-Criterion Optimization. EMO 2025. Lecture Notes in Computer Science, vol 15512. Springer, Singapore. https://doi.org/10.1007/978-981-96-3506-1_8

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  • DOI: https://doi.org/10.1007/978-981-96-3506-1_8

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