  Bayesian Optimization (BO) is used to find the global optima of black box
functions. In this work, we propose a practical BO method of function
compositions where the form of the composition is known but the constituent
functions are expensive to evaluate. By assuming an independent Gaussian
process (GP) model for each of the constituent black-box function, we propose
Expected Improvement (EI) and Upper Confidence Bound (UCB) based BO algorithms
and demonstrate their ability to outperform not just vanilla BO but also the
current state-of-art algorithms. We demonstrate a novel application of the
proposed methods to dynamic pricing in revenue management when the underlying
demand function is expensive to evaluate.
