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We consider the standard optimistic bilevel optimization problem, inparticular upper- and lower-level constraints can be coupled. By means of thelower-level value function, the problem is transformed into a single-leveloptimization problem with a penalization of the value function constraint. Fortreating the latter problem, we develop a framework that does not rely on thedirect computation of the lower-level value function or its derivatives. Foreach penalty parameter, the framework leads to a semismooth system ofequations. This allows us to extend the semismooth Newton method to bileveloptimization. Besides global convergence properties of the method, we focus onachieving local superlinear convergence to a solution of the semismooth system.To this end, we formulate an appropriate CD-regularity assumption and derivesuffcient conditions so that it is fulfilled. Moreover, we develop conditionsto guarantee that a solution of the semismooth system is a local solution ofthe bilevel optimization problem. Extensive numerical experiments on $124$examples of nonlinear bilevel optimization problems from the literature showthat this approach exhibits a remarkable performance, where only a few penaltyparameters need to be considered.

Semismooth Newton-type method for bilevel optimization: Global convergence and extensive numerical experiments