Approximation and exact penalization in simple bilevel variational problems

A simple bilevel variational problem, where the lower level is a variational inequality while the upper level is an optimization problem, is studied. We consider an inexact version of the lower problem, which guarantees enough regularity to allow the exploitation of techniques of exact penalization. Moreover, cutting planes are used to approximate the Minty gap function of the lower level. Algorithms to solve the resulting inexact bilevel problem are devised relying on these techniques and approximations. Finally, their convergence is studied in detail by analyzing the effect of the given inexactness.