A Sequential Algorithm for Personalized Optimization of Noisy Black-Box Functions

Personalized optimization is a new strategy for optimization problems with environmental variables. For noisy black-box functions, this paper proposes a sequential algorithm to implement personalized optimization. We first use an initial design to build a noisy Gaussian process model as the statistical surrogate model of the black-box function. Based on the surrogate model, we introduce a new acquisition function, called hierarchical expected quantile improvement (HEQI), and give the next trial site of the control variables through maximizing HEQI. The next trial site of the environmental variables is obtained via a space-filling criterion. Conducting the above steps sequentially, we can approach the solution of the personalized optimization problem. A number of numerical experiments indicate that, the HEQI-based personalized optimization algorithm outperforms those based on other Bayesian optimization strategies at various levels of noises.