Optimization Under Unknown Constraints

Optimization of complex functions, such as the output of computer simulators, is a difficult task that has received much attention in the literature. A less studied problem is that of optimization under unknown constraints, i.e., when the simulator must be invoked both to determine the typical real-valued response and to determine if a constraint has been violated, either for physical or policy reasons. We develop a statistical approach based on Gaussian processes and Bayesian learning to both approximate the unknown function and estimate the probability of meeting the constraints. A new integrated improvement criterion is proposed to recognize that responses from inputs that violate the constraint may still be informative about the function, and thus could potentially be useful in the optimization. The new criterion is illustrated on synthetic data, and on a motivating optimization problem from health care policy.