Journal
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
Volume 173, Issue 1, Pages 183-202Publisher
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10957-017-1085-4
Keywords
Pessimistic bilevel problem; Viscosity solution; Lower semicontinuous set-valued map; Security value
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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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