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Numerically tractable optimistic bilevel problems

Author

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  • Lorenzo Lampariello

    (Roma Tre University)

  • Simone Sagratella

    (Sapienza University of Rome)

Abstract

We consider a class of optimistic bilevel problems. Specifically, we address bilevel problems in which at the lower level the objective function is fully convex and the feasible set does not depend on the upper level variables. We show that this nontrivial class of mathematical programs is sufficiently broad to encompass significant real-world applications and proves to be numerically tractable. From this respect, we establish that the stationary points for a relaxation of the original problem can be obtained addressing a suitable generalized Nash equilibrium problem. The latter game is proven to be convex and with a nonempty solution set. Leveraging this correspondence, we provide a provably convergent, easily implementable scheme to calculate stationary points of the relaxed bilevel program. As witnessed by some numerical experiments on an application in economics, this algorithm turns out to be numerically viable also for big dimensional problems.

Suggested Citation

  • Lorenzo Lampariello & Simone Sagratella, 2020. "Numerically tractable optimistic bilevel problems," Computational Optimization and Applications, Springer, vol. 76(2), pages 277-303, June.
    • Handle: RePEc:spr:coopap:v:76:y:2020:i:2:d:10.1007_s10589-020-00178-y
    DOI: 10.1007/s10589-020-00178-y
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    10. Simone Sagratella, 2017. "Algorithms for generalized potential games with mixed-integer variables," Computational Optimization and Applications, Springer, vol. 68(3), pages 689-717, December.
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    Cited by:

    1. Alain B. Zemkoho & Shenglong Zhou, 2021. "Theoretical and numerical comparison of the Karush–Kuhn–Tucker and value function reformulations in bilevel optimization," Computational Optimization and Applications, Springer, vol. 78(2), pages 625-674, March.
    2. Francesco Cesarone & Lorenzo Lampariello & Davide Merolla & Jacopo Maria Ricci & Simone Sagratella & Valerio Giuseppe Sasso, 2023. "A bilevel approach to ESG multi-portfolio selection," Computational Management Science, Springer, vol. 20(1), pages 1-23, December.
    3. Jörg Fliege & Andrey Tin & Alain Zemkoho, 2021. "Gauss–Newton-type methods for bilevel optimization," Computational Optimization and Applications, Springer, vol. 78(3), pages 793-824, April.
    4. Lorenzo Lampariello & Christoph Neumann & Jacopo M. Ricci & Simone Sagratella & Oliver Stein, 2020. "An explicit Tikhonov algorithm for nested variational inequalities," Computational Optimization and Applications, Springer, vol. 77(2), pages 335-350, November.
    5. Lampariello, Lorenzo & Neumann, Christoph & Ricci, Jacopo M. & Sagratella, Simone & Stein, Oliver, 2021. "Equilibrium selection for multi-portfolio optimization," European Journal of Operational Research, Elsevier, vol. 295(1), pages 363-373.
    6. Lorenzo Lampariello & Gianluca Priori & Simone Sagratella, 2022. "On the solution of monotone nested variational inequalities," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 96(3), pages 421-446, December.

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