Many simulation-based engineering optimization problems involve hidden constraints that are only detected when the simulator that calculates function values fails at a given input vector. However, relatively few papers in the literature address this important issue in real-world optimization problems. This paper proposes a general approach for handling hidden constraints in surrogate-based constrained black-box optimization that imputes on the objective and constraint function values whenever a failed simulation point is encountered in the search for a global optimum. The main idea is to impute an objective or constraint function value that is worse than the worst value among neighbor sample points within a certain radius of the failed simulation point. This imputation method is compared with the common approach of imputing a large function value at failed simulation points when these methods are used with the COBRA and ConstrLMSRBF algorithms on a 31-dimensional supersonic business jet optimization problem with 78 black-box inequality constraints. The COBRA and ConstrLMSRBF algorithms with the proposed imputation method yield substantial improvements over an initial feasible point in multiple trials. Moreover, the proposed method obtained better results than the alternative imputation method when they are used with the COBRA and ConstrLMSRBF algorithms on the business jet problem. This suggests that the proposed imputation idea is a promising approach for surrogate-based optimization. Additionally, COBRA with a local search strategy yielded the best improvements on the business jet application when the computational budget is limited to 1000 function evaluations.