@misc{MehlitzMinchenkoZemkoho, author = {Mehlitz, Patrick and Minchenko, Leonid I. and Zemkoho, Alain B.}, title = {A note on partial calmness for bilevel optimization problems with linearly structured lower level}, series = {Optimization Letters}, journal = {Optimization Letters}, issn = {1862-4480}, doi = {10.1007/s11590-020-01636-6}, pages = {15}, abstract = {Partial calmness is a celebrated but restrictive property of bilevel optimization problems whose presence opens a way to the derivation of Karush-Kuhn-Tucker-type necessary optimality conditions in order to characterize local minimizers. In the past, sufficient conditions for the validity of partial calmness have been investigated. In this regard, the presence of a linearly structured lower level problem has turned out to be beneficial. However, the associated literature suffers from inaccurate results. In this note, we clarify some regarding erroneous statements and visualize the underlying issues with the aid of illustrative counterexamples.}, language = {en} } @misc{MehlitzMinchenko, author = {Mehlitz, Patrick and Minchenko, Leonid I.}, title = {R-Regularity of Set-Valued Mappings Under the Relaxed Constant Positive Linear Dependence Constraint Qualification with Applications to Parametric and Bilevel Optimization}, series = {Set-Valued and Variational Analysis}, volume = {30}, journal = {Set-Valued and Variational Analysis}, number = {1}, issn = {1877-0541}, doi = {10.1007/s11228-021-00578-0}, pages = {179 -- 205}, abstract = {The presence of Lipschitzian properties for solution mappings associated with nonlinear parametric optimization problems is desirable in the context of, e.g., stability analysis or bilevel optimization. An example of such a Lipschitzian property for set-valued mappings, whose graph is the solution set of a system of nonlinear inequalities and equations, is R-regularity. Based on the so-called relaxed constant positive linear dependence constraint qualification, we provide a criterion ensuring the presence of the R-regularity property. In this regard, our analysis generalizes earlier results of that type which exploited the stronger Mangasarian-Fromovitz or constant rank constraint qualification. Afterwards, we apply our findings in order to derive new sufficient conditions which guarantee the presence of R-regularity for solution mappings in parametric optimization. Finally, our results are used to derive an existence criterion for solutions in pessimistic bilevel optimization and a sufficient condition for the presence of the so-called partial calmness property in optimistic bilevel optimization.}, language = {en} }