Black box functions, such as computer experiments, often have multiple optima over the input space of the objective function. While traditional optimization routines focus on finding a single best optimum, we sometimes want to consider the relative merits of multiple optima. First we need a search algorithm that can identify multiple local optima. Then we consider that blindly choosing the global optimum may not always be best. In some cases, the global optimum may not be robust to small deviations in the inputs, which could lead to output values far from the optimum. In those cases, it would be better to choose a slightly less extreme optimum that allows for input deviation with small change in the output; such an optimum would be considered more robust. We use a Bayesian decision theoretic approach to develop a utility function for selecting among multiple optima.
The black box functions found in computer experiments often result in multimodal optimization programs. Optimization that focuses on a single best optimum may not achieve the goal of getting the best answer for the purposes of the experiment. This paper builds upon an algorithm introduced in [1] that is successful for finding multiple optima within the input space of the objective function. Here we introduce an alternative cluster search algorithm for finding these optima, making use of clustering. The cluster search algorithm has several advantages over the earlier algorithm. It gives a forward view of the optima that are present in the input space so the user has a preview of what to expect as the optimization process continues. It employs pattern search, in many instances, closer to the minimum's location in input space, saving on simulator point computations. At termination, this algorithm does not need additional verification that a minimum is a duplicate of a previously found minimum, which also saves on simulator point computations. Finally, it finds minima that can be "hidden" by close larger minima.
Global sensitivity analysis provides information on the relative importance of the input variables for simulator functions used in computer experiments. It is more conclusive than screening methods for determining if a variable is influential, especially if a variable's influence is derived from its interactions with other variables. In this paper, we develop a method for providing global sensitivities with estimated accuracy. A treed Gaussian process serves as a statistical emulator of the black box function. A sequential experimental design makes effective and efficient use of simulator evaluations by adaptively sampling points that are expected to provide the maximum improvement to the emulator model. The method accounts for both sampling error and emulator error.
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