# Deterministic Global Robust Optimization using McCormick Relaxations

Markus Beckers^{*} and Uwe Naumann

This talk describes an algorithm for robust optimization consisting of two main independently developed components. The algorithm combines uncertainty quantification and global optimization (minimization) to a robust optimization. First uncertainty quantification is used to produce a robust version of the functional $F: R^{n} \rightarrow R$. Subsequently a McCormick based branch and bound optimization approach is applied to this robust function. The uncertainty quantification is based on moments method. Moments method assumes the inputs of the underlying vector function to be probabilistic because of possible measurement errors. Therefore the mean and variance of the outputs are approximated by a Taylor series. Algorithmic Differentiation (AD) is employed to compute the required derivatives either in tangent-linear or adjoint mode. The second component is a deterministic branch and bound algorithm for global optimization based on McCormick relaxations. McCormick relaxations are convex underestimators of factorable functions which are possibly nonsmooth. Their subgradients are used instead of interval analysis in the branch and bound method. The fact that the computation of subgradients of McCormick relaxations is equivalent to AD was shown in previous work. The combination of these two procedures ensures a more stable minimum accounting the possible uncertainties in the inputs.

Mathematics Subject Classification: 99X99

Keywords: Global Optimization; Robust Design; McCormick Relaxations; Algorithmic Differentiation; Uncertainty Quantification

Minisymposion: Nonsmooth Optimization