(456a) Uncertainty Considerations for Surrogate Functions for Constrained Grey-Box Optimization
One of the main components of any model-based derivative-free or black-box optimization method, is the identification of good surrogate approximations. This remains to be one of the major challenges of optimization without analytical forms or accurate derivatives, since the uncertainty caused by the fitted parameters of the surrogate models, as well as the mismatch between the underlying unknown function and the fitted model, highly affect the performance and reliability of any algorithm. Specifically, it has been observed that sampling plays a major role in the identification of the optimal parameters, since given a different set of samples, the optimal parameters of a surrogate model can be different. Moreover, the actual form of the input-output model is unknown, magnifying the observed variability in the surrogate model parameters given a different set of samples. It is important to develop surrogate models which provide good search directions, within which the surrogate models will be refined. Thus, the main challenge becomes the identification of promising refined search spaces, when the surrogate models cannot be entirely trusted as exact representations of the unknown functions.
In this work, we investigate the effects of surrogate parameter uncertainty caused by two main sources: (a) sampling of the search space and (b) selection of the appropriate surrogate function. We show that through the use of concepts from robust counterpart optimization [5-6], we can formulate a 'best-case' constrained grey-box representation, as well as a 'worst-case' constrained grey-box representation, which can serve as probabilistic lower and upper bounds on the actual optimum, respectively. We will show this concept through several examples using a set of different surrogate function types (i.e., linear, polynomial, kriging and radial basis functions). We will also discuss how these uncertainty-based bounds are incorporated within our constrained grey-box optimization framework to improve the reliability by mitigating the variability caused by the initial sampling stage. We will show results of our improved ARGONAUT framework which incorporates all of the above, as well as improved variable and term selection methods. Finally, we show that through efficient parallelization of the sampling and model identification stages, we can solve problems with large dimensionality and total number of constraints with reduced computational running time.
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