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We propose an optimal numerical test for genuine multipartite nonlocality based on linear programming. In particular, we consider two non-equivalent models of local hidden variables, namely the Svetlichny and the no-signaling bilocal model. While our knowledge concerning these models is well established for Bell scenarios involving two measurement settings per party, the general case based on an arbitrary number of settings is a considerably more challenging task and very little work has been done in this field. In this paper, we applied such general tests to detect and characterize genuine $n$-way nonlocal correlations for various states of three qubits and qutrits. As a measure of nonlocality, we use the probability of violation of local realism under randomly sampled observables, and the strength of nonlocality, described by the resistance to white noise admixture. In particular, we analyze to what extent the Bell scenario involving two measurement settings can be used to determine genuine $n$-way non-local correlations generated for more general models. In addition, we propose a simple procedure to detect genuine multipartite nonlocality for randomly chosen settings with up to 100% efficiency.