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Bayesian optimization (BO) is a powerful class of data-driven techniques for the maximization of expensive-to-evaluate objective functions. These techniques construct a Gaussian process (GP) regression for representing the objective function according to the latest available function evaluations and sequentially select samples and evaluate the function by maximizing an acquisition function. The primary assumption in most BO policies is that the objective function has a uniform level of smoothness over the input space, modeled by a kernel function. However, the uniform smoothness assumption is likely to be violated in a wide range of practical problems, primary domains in which the objective function is evaluated differently at various regions of input space (e.g., through different experiments, software, or approximators). This article develops a BO framework capable of optimizing expensive smooth-varying …