Bayesian Optimization with Unknown Constraints

Bayesian optimization (Mockus et al., 1978) is a method for performing global optimization of unknown "black box" objectives that is particularly appropriate when objective function evaluations are expensive (in any sense, such as time or money). For example, consider a food company trying to design a low-calorie variant of a popular cookie. In this case, the design space is the space of possible recipes and might include several key parameters such as quantities of various ingredients and baking times. Each evaluation of a recipe entails computing (or perhaps actually measuring) the number of calories in the proposed cookie. Bayesian optimization can be used to propose new candidate recipes such that good results are found with few evaluations. Now suppose the company also wants to ensure the taste of the cookie is not compromised when calories are reduced. Therefore, for each proposed low-calorie recipe, they perform a taste test with sample customers. Because different people, or the same people at different times, have differing opinions about the taste of cookies, the company decides to require that at least 95% of test subjects must like the new cookie.