Robust optimization for long running computer simulations with a focus on robustness measures

Robust optimization determines how the input variables dispersion is propagated on the output variables. This is of great practical relevance: For example, the quality of a product is influenced decisively by production tolerances. In industrial applications it is important to characterize the range of variation with appropriate measures. The industry is particularly interested in accurate limits of the output distribution or its centered part. In this article the mathematical characterization of robustness is discussed under the viewpoint of its practical applicability. It is shown, that the usually used robustness measures mean for central tendency and standard deviation for dispersion produce inaccurate limits. Instead several measures based on quantiles are proposed. The median is used as measure of central tendency while different quantile ranges are used as measure of dispersion. They are compared for the robust optimization of mathematical functions and industria l applications. The advantages of the quantile measures are pointed out. The computation of quantiles is expensive, because it needs many function evaluations. Due to their long runtime only a few simulations can be executed in practice. A methodology tailored to this situation is proposed. It is based on the use of metamodels. Starting with a few real simulations a metamodel for the system is build. Further ones for median and dispersion of the output variables are derived from it. This enables the user to perform a full multicriteria robust optimization on the whole parameter range. The methodology takes the tolerances of the metamodels into account. A new measure for the tolerance of metamodels which are derived from metamodels is presented. It is used to estimate the accuracy of the quantile models. The tolerance can easily be integrated in the robust optimization process.