Gradients of Acquisition Functions for Bi-objective Bayesian optimization

Multi-objective Bayesian optimization (MOBO) searches for optimal solutions by maximizing acquisition functions to optimize expensive black-box functions globally. MOBO frequently employs the probability of improvement (POI), expected hypervolume improvement (EHVI), and truncated expected hypervolume improvement (TEHVI) acquisition functions. Based on the POI, this paper proposes the truncated probability of improvement (TPoI) that leverages prior knowledge of objective values via the truncated normal distribution. Additionally, this paper proposes explicit formulas for computing the gradient of POI, the gradient of TPoI, and the gradient of TEHVI.