Optimizations Under Uncertainty Using Gradients, Hessians, and Surrogate Models

Abstract

In this article, a first-order moment method and a kriging surrogate model are used for optimizations under uncertainty applied to two-bar truss designs and two-dimensional lift-constrained drag minimizations. Given uncertainties in statistically independent, random, normally distributed input variables, the two approaches are used to propagate these uncertainties through the mathematical model and to approximate output statistics of interest. To assess the validity of the approximations, the results are compared with full Monte Carlo simulations. When using first-order moment methods for robust optimizations, first-order sensitivity derivatives appear in the objective function and system constraints. Therefore, second-order sensitivity derivatives are needed for gradient-based optimization approaches. When the kriging surrogate model is used to calculate the objective function value and system constraints, it will be shown that a gradient predictor for the kriging model can be successfully used for gradient-based optimizations. Both the kriging and first-order moment method approaches enable predictions of the mean and variance of quantities of interest while at the same time keeping the computational cost for optimization under uncertainty problems manageable. The novelty of this article is the use of a kriging surrogate model for uncertainty propagation in a gradient-based robust optimization framework.