Any performance analysis based on stochastic simulation is subject to the errors inherent in misspecifying the modeling assumptions, particularly the input distributions. In situations with little support from data, we investigate the use of worst-case analysis to analyze these errors, by representing the partial, nonparametric knowledge of the input models via optimization constraints. We study the performance and robustness guarantees of this approach. We design and analyze a numerical scheme for solving a general class of simulation objectives and uncertainty specifications. The key steps involve a randomized discretization of the probability spaces, a simulable unbiased gradient estimator using a nonparametric analog of the likelihood ratio method, and a Frank-Wolfe (FW) variant of the stochastic approximation (SA) method (which we call FWSA) run on the space of input probability distributions. A convergence analysis for FWSA on nonconvex problems is provided. We test the performance of our approach via several numerical examples.