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Inner Regularizations and Viscosity Solutions for Pessimistic Bilevel Optimization Problems

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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.

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Acknowledgements

The authors would like to thank the Associate Editor and the anonymous referees for their valuable comments and suggestions that have significantly improved an earlier version of the paper.

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Authors and Affiliations

  1. Department of Mathematics and Applications R. Caccioppoli, University of Naples Federico II, Via Claudio 21, 80125, Napoli, Italy

    M. Beatrice Lignola

  2. Department of Economics and Statistics & Centre for Studies in Economics and Finance, University of Naples Federico II, Via Cinthia, 80126, Napoli, Italy

    Jacqueline Morgan

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  1. M. Beatrice Lignola
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  2. Jacqueline Morgan
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Correspondence to Jacqueline Morgan.

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Communicated by Alexander Mitsos.

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Lignola, M.B., Morgan, J. Inner Regularizations and Viscosity Solutions for Pessimistic Bilevel Optimization Problems. J Optim Theory Appl 173, 183–202 (2017). https://doi.org/10.1007/s10957-017-1085-4

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