  In this paper we consider the problem of finding stable maxima of expensive
(to evaluate) functions. We are motivated by the optimisation of physical and
industrial processes where, for some input ranges, small and unavoidable
variations in inputs lead to unacceptably large variation in outputs. Our
approach uses multiple gradient Gaussian Process models to estimate the
probability that worst-case output variation for specified input perturbation
exceeded the desired maxima, and these probabilities are then used to (a) guide
the optimisation process toward solutions satisfying our stability criteria and
(b) post-filter results to find the best stable solution. We exhibit our
algorithm on synthetic and real-world problems and demonstrate that it is able
to effectively find stable maxima.
