Inner Regularizations and Viscosity Solutions for Pessimistic Bilevel Optimization Problems

M. Beatrice Lignola () and Jacqueline Morgan
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M. Beatrice Lignola: University of Naples Federico II

Journal of Optimization Theory and Applications, 2017, vol. 173, issue 1, No 9, 183-202

Abstract: 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.

Keywords: Pessimistic bilevel problem; Viscosity solution; Lower semicontinuous set-valued map; Security value; 49J45; 49J53; 90C47 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (6)

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Working Paper: Inner Regularizations and Viscosity Solutions for Pessimistic Bilevel Optimization Problems (2016)
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DOI: 10.1007/s10957-017-1085-4

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