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Solving integer indefinite quadratic bilevel programs with multiple objectives at the upper level

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Abstract

Bilevel programming is characterized by the existence of two optimization problems in which the constraint region of the upper level problem is implicitly determined by the lower level optimization problem. This hierarchical design of optimization is suitable to model a large number of real-life applications. However, when dealing with a non linear multi-objective optimization context, new complexities arise due to conflicting objectives. In this paper, an exact method is described to solve an integer indefinite quadratic bilevel maximization problem with multiple objectives at the upper level, where the objective functions at both levels are the product of two linear functions. The algorithm suggested aims to produce a set of efficient solutions by employing a branch and cut approach. It optimizes the indefinite quadratic problem of the upper level within the feasible region of the original problem in an iterative manner. Then, it introduces the Dantzig cut technique to identify the optimal solution for the integer indefinite quadratic bilevel programming problem. Additionally, the algorithm utilizes an efficient cut that reduces the search process for obtaining the set of efficient solutions of the main problem, along with a branching constraint for the integer decision variables. The algorithm was implemented and tested on instances generated randomly, yielding positive outcomes.

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Acknowledgements

The Authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

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Authors and Affiliations

  1. LaROMaD Laboratory, Department of Operations Research, Faculty of Mathematics, USTHB, P.B. 32 El-Alia, 16111, Bab Ezzouar, Algiers, Algeria

    Fatima Fali, Yasmin Cherfaoui & Mustapha Moulaï

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  1. Fatima Fali
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  2. Yasmin Cherfaoui
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  3. Mustapha Moulaï
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Correspondence to Fatima Fali.

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Fali, F., Cherfaoui, Y. & Moulaï, M. Solving integer indefinite quadratic bilevel programs with multiple objectives at the upper level. J. Appl. Math. Comput. 70, 1153–1170 (2024). https://doi.org/10.1007/s12190-023-01968-3

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