This study addresses the problem of minimizing functions whose values are obtained by running an expensive computer simulation and where the simulation may fail to converge for some points, i.e. the objective function is discontinuous. An algorithm is proposed which is comprised of three major steps: generating a global surrogate model of the objective function; using an evolutionary algorithm to identify regions where minimizers are likely to exist and then using an efficient local search to converge to a minimizer. In order to handle discontinuity points we implement two mechanisms: a) a global penalty function is generated by interpolation and is used to modify the global surrogate model such that the function value predicted by the global surrogate model are increased in the vicinity of discontinuity points b) in case a discontinuity point is encountered during the local search the search region is temporarily reduced so as to seek a valid point in a smaller region around the current valid point. Performance analysis from minimization of mathematical test functions and a real-world application of airfoil shape optimization show the proposed algorithm efficiently minimizes expensive black-box functions with discontinuity points.