  In this paper, we consider mixed-integer nonsmooth constrained optimization
problems whose objective/constraint functions are available only as the output
of a black-box zeroth-order oracle (i.e., an oracle that does not provide
derivative information) and we propose a new derivative-free linesearch-based
algorithmic framework to suitably handle those problems. We first describe a
scheme for bound constrained problems that combines a dense sequence of
directions (to handle the nonsmoothness of the objective function) with
primitive directions (to handle discrete variables). Then, we embed an exact
penalty approach in the scheme to suitably manage nonlinear (possibly
nonsmooth) constraints. We analyze the global convergence properties of the
proposed algorithms toward stationary points and we report the results of an
extensive numerical experience on a set of mixed-integer test problems.
