Setting targets for surrogate-based optimization

In the context of surrogate-based optimization (SBO), most designers have still very little guidance on when to stop and how to use infill measures with target requirements (e.g., one-stage approach for goal seeking and optimization); the reason: optimum estimates independent of the surrogate and optimization strategy are seldom available. Hence, optimization cycles are typically stopped when resources run out (e.g., number of objective function evaluations/time) or convergence is perceived, and targets are empirically set which may affect the effectiveness and efficiency of the SBO approach. This work presents an approach for estimating the minimum (target) of the objective function using concepts from extreme order statistics which relies only on the training data (sample) outputs. It is assumed that the sample inputs are randomly distributed so the outputs can be considered a random variable, whose density function is bounded (a, b), with the minimum (a) as its lower bound. Specifically, an estimate of the minimum (a) is obtained by: (i) computing the bounds (using training data and the moment matching method) of a selected set of analytical density functions (catalog), and (ii) identifying the density function in the catalog with the best match to the sample outputs distribution and corresponding minimum estimate (a). The proposed approach makes no assumption about the nature of the objective functions, and can be used with any surrogate, and optimization strategy even with high dimensional problems. The effectiveness of the proposed approach was evaluated using a compact catalog of Generalized Beta density functions and well-known analytical optimization test functions, i.e., F2, Hartmann 6D, and Griewangk 10D and in the optimization of a field scale alkali-surfactant-polymer enhanced oil recovery process. The results revealed that: (a) the density function (from a catalog) with the best match to a function outputs distribution, was the same for both large and reduced samples, (b) the true optimum value was always within a 95% confidence interval of the estimated minimum distribution, and (c) the estimated minimum represents a significant improvement over the present best solution and an excellent approximation of the true optimum value.