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Inner Regularizations and Viscosity Solutions for Pessimistic Bilevel Optimization Problems

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  • M. Beatrice Lignola

    (University of Naples Federico II)

  • Jacqueline Morgan

    (University of Naples Federico II)

Abstract

Pessimistic bilevel optimization problems are not guaranteed to have a solution even when restricted classes of data are involved. Thus, we propose a concept of viscosity solution, which satisfactorily obviates the lack of optimal solutions since it allows to achieve in appropriate conditions the security value. Differently from the viscosity solution concept for optimization problems, introduced by Attouch (SIAM J Optim 6:769–806, 1996) and defined through a viscosity function that aims at regularizing the objective function, viscosity solutions for pessimistic bilevel optimization problems are defined through regularization families of the solutions map to the lower-level optimization. These families are termed “inner regularizations” since they approach the optimal solutions map from the inside. First, we investigate, in Banach spaces, several classical regularizations of parametric constrained minimum problems giving sufficient conditions for getting inner regularizations; then, we establish existence results for the corresponding viscosity solutions under possibly discontinuous data.

Suggested Citation

  • M. Beatrice Lignola & Jacqueline Morgan, 2017. "Inner Regularizations and Viscosity Solutions for Pessimistic Bilevel Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 173(1), pages 183-202, April.
  • Handle: RePEc:spr:joptap:v:173:y:2017:i:1:d:10.1007_s10957-017-1085-4
    DOI: 10.1007/s10957-017-1085-4
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    References listed on IDEAS

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    1. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(6), pages 1157-1160, December.
    2. M. Beatrice Lignola & Jacqueline Morgan, 2014. "Viscosity Solutions for Bilevel Problems with Nash Equilibrium Constraints," CSEF Working Papers 367, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy, revised 02 Oct 2014.
    3. Stein, Oliver & Still, Georg, 2002. "On generalized semi-infinite optimization and bilevel optimization," European Journal of Operational Research, Elsevier, vol. 142(3), pages 444-462, November.
    4. M. Beatrice Lignola & Jacqueline Morgan, 2012. "Approximating Security Values of MinSup Problems with Quasi-variational Inequality Constraints," CSEF Working Papers 321, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy, revised 09 Oct 2014.
    5. Henry Bonnel & Jacqueline Morgan, 2011. "Optimality Conditions for Semivectorial Bilevel Convex Optimal Control Problems," CSEF Working Papers 301, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy.
    6. Mengwei Xu & Jane Ye, 2014. "A smoothing augmented Lagrangian method for solving simple bilevel programs," Computational Optimization and Applications, Springer, vol. 59(1), pages 353-377, October.
    7. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(5), pages 1025-1031, October.
    8. Harald Günzel & Hubertus Jongen & Oliver Stein, 2007. "On the closure of the feasible set in generalized semi-infinite programming," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 15(3), pages 271-280, September.
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    Cited by:

    1. Francesco Caruso & M. Beatrice Lignola & Jacqueline Morgan, 2020. "Regularization and Approximation Methods in Stackelberg Games and Bilevel Optimization," Springer Optimization and Its Applications, in: Stephan Dempe & Alain Zemkoho (ed.), Bilevel Optimization, chapter 0, pages 77-138, Springer.
    2. 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.
    3. Sjur Didrik Flåm, 2021. "Games and cost of change," Annals of Operations Research, Springer, vol. 301(1), pages 107-119, June.
    4. M. Beatrice Lignola & Jacqueline Morgan, 2019. "Further on Inner Regularizations in Bilevel Optimization," Journal of Optimization Theory and Applications, Springer, vol. 180(3), pages 1087-1097, March.
    5. Francesco Caruso & Maria Carmela Ceparano & Jacqueline Morgan, 2019. "Subgame Perfect Nash Equilibrium: A Learning Approach via Costs to Move," Dynamic Games and Applications, Springer, vol. 9(2), pages 416-432, June.
    6. Francesco Caruso & Maria Carmela Ceparano & Jacqueline Morgan, 2022. "Asymptotic Behavior of Subgame Perfect Nash Equilibria in Stackelberg Games," CSEF Working Papers 661, Centre for Studies in Economics and Finance (CSEF), University of Naples, Italy.

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