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Asymptotic stationarity and regularity conditions turned out to be quiteuseful to study the qualitative properties of numerical solution methods forstandard nonlinear and complementarity-constrained programs. In this paper, wefirst extend these notions to nonlinear optimization problems with nonsmoothbut Lipschitzian data functions in order to find reasonable notions ofasymptotic stationarity and regularity in terms of Clarke's and Mordukhovich'ssubdifferential construction. Particularly, we compare the associated novelasymptotic constraint qualifications with already existing ones. The secondpart of the paper presents two applications of the obtained theory. On the onehand, we specify our findings for complementarity-constrained optimizationproblems and recover recent results from the literature which demonstrates thepower of the approach. Furthermore, we hint at potential extensions to or- andvanishing-constrained optimization. On the other hand, we demonstrate theusefulness of asymptotic regularity in the context of bilevel optimization.More precisely, we justify a well-known stationarity system for affinelyconstrained bilevel optimization problems in a novel way. Afterwards, wesuggest a solution algorithm for this class of bilevel optimization problemswhich combines a penalty method with ideas from DC-programming. After a briefconvergence analysis, we present results of some numerical experiments.

Asymptotic regularity for Lipschitzian nonlinear optimization problems with applications to complementarity-constrained and bilevel programming