  In this paper, we revisit the classical problem of solving over-determined
systems of nonsmooth equations numerically. We suggest a nonsmooth
Levenberg--Marquardt method for its solution which, in contrast to the existing
literature, does not require local Lipschitzness of the data functions. This is
possible when using Newton-differentiability instead of semismoothness as the
underlying tool of generalized differentiation. Conditions for fast local
convergence of the method are given. Afterwards, in the context of
over-determined mixed nonlinear complementarity systems, our findings are
applied, and globalized solution methods, based on a residual induced by the
maximum and the Fischer--Burmeister function, respectively, are constructed.
The assumptions for fast local convergence are worked out and compared.
Finally, these methods are applied for the numerical solution of bilevel
optimization problems. We recall the derivation of a stationarity condition
taking the shape of an over-determined mixed nonlinear complementarity system
involving a penalty parameter, formulate assumptions for local fast convergence
of our solution methods explicitly, and present results of numerical
experiments. Particularly, we investigate whether the treatment of the
appearing penalty parameter as an additional variable is beneficial or not.

Optimization and Control