  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.
