Nonconvex optimization problems involving both continuous and discrete variables remain a theoretical challenge with important practical implications. Recently, Bayesian optimization algorithms have been extended to problems with mixed variables for which they offer a mathematical framework. Bayesian algorithms contain an internal optimization problem, typically the maximization of the expected improvement. Although a large number of evaluations is possible for this internal problem, it still contains mixed variables. In this work, we propose to address such problems by composing state-of-the-art continuous and discrete global optimization algorithms. These algorithms are tested on a series of analytical and expected improvement functions. It turns out that the best algorithms for composition are the noisy variants, i.e., the algorithms that take into account the possibility of an inaccurate evaluation of the search points. It is also found that accurately optimizing the expected improvement is important for the efficiency of the Bayesian search.