Sampling Criteria for Constrained Bayesian Optimization under Uncertainty

Abstract : We consider the problem of chance constrained optimization where the objective and the constraint functions are affected by uncertainties and are computationally costly. Bayesian optimization is an appropriate family of methods to address such problems. We first propose a two-step acquisition criterion defined in the joint space of optimization variables and uncertain parameters. The objective and the constraints are aggregated through a feasible improvement measure and the two steps consist in the optimization of the expectation and the one-step-ahead variance of this criterion. To ease the computational burden, an analytical approximation to the one-step-ahead variance is proposed. Additionally, we also account for the possible correlation between the constraints. This is done by considering a vector-valued "input as output" joined Gaussian process which improves the constraints modeling accuracy and consequently the optimization procedure. The correlations between the constraints are further exploited by allowing each constraint to be evaluated for different uncertain parameters and by optimally selecting a subset of constraints to be evaluated at each iteration, thus avoiding unnecessary computations. Numerical tests confirm the applicability and potential gains brought by these methods, such a faster convergence speed and better scaling with respect to the number of constraints if compared to alternative optimization methods.