  The Inexact Restoration approach has proved to be an adequate tool for
handling the problem of minimizing an expensive function within an arbitrary
feasible set by using different degrees of precision in the objective function.
The Inexact Restoration framework allows one to obtain suitable convergence and
complexity results for an approach that rationally combines low- and
high-precision evaluations. In the present research, it is recognized that many
problems with expensive objective functions are nonsmooth and, sometimes, even
discontinuous. Having this in mind, the Inexact Restoration approach is
extended to the nonsmooth or discontinuous case. Although optimization phases
that rely on smoothness cannot be used in this case, basic convergence and
complexity results are recovered. A derivative-free optimization phase is
defined and the subproblems that arise at this phase are solved using a
regularization approach that take advantage of different notions of
stationarity. The new methodology is applied to the problem of reproducing a
controlled experiment that mimics the failure of a dam.
