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In this paper, mixed-integer nonsmooth constrained optimization problems are considered, where objective/constraint functions are available only as the output of a black-box zeroth-order oracle that does not provide derivative information. A new derivative-free linesearch-based algorithmic framework is proposed to suitably handle those problems. First, 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 is described. Then, an exact penalty approach is embedded in the scheme to suitably manage nonlinear (possibly nonsmooth) constraints. Global convergence properties of the proposed algorithms toward stationary points are analyzed and results of an extensive numerical experience on a set of mixed-integer test problems are reported. [ABSTRACT FROM AUTHOR]