  We consider the standard optimistic bilevel optimization problem, in
particular upper- and lower-level constraints can be coupled. By means of the
lower-level value function, the problem is transformed into a single-level
optimization problem with a penalization of the value function constraint. For
treating the latter problem, we develop a framework that does not rely on the
direct computation of the lower-level value function or its derivatives. For
each penalty parameter, the framework leads to a semismooth system of
equations. This allows us to extend the semismooth Newton method to bilevel
optimization. Besides global convergence properties of the method, we focus on
achieving local superlinear convergence to a solution of the semismooth system.
To this end, we formulate an appropriate CD-regularity assumption and derive
suffcient conditions so that it is fulfilled. Moreover, we develop conditions
to guarantee that a solution of the semismooth system is a local solution of
the bilevel optimization problem. Extensive numerical experiments on $124$
examples of nonlinear bilevel optimization problems from the literature show
that this approach exhibits a remarkable performance, where only a few penalty
parameters need to be considered.
