Uncertainty Quantification in Optimization

Abstract

We consider constrained optimization problems affected by uncertainty, where the objective function or the restrictions involve random variables $\varvec{u}$. In this situation, the solution of the optimization problem is a random variable $\varvec{x}\left( \varvec{u} \right)$: we are interested in the determination of its distribution of probability. By using Uncertainty Quantification approaches, we may find an expansion of $\varvec{x}\left( \varvec{u} \right)$ in terms of a Hilbert basis ${\mathcal{F}} = \left\{ {\varphi_{i} :i \in {\mathbb{N}}^{*} } \right\}$. We present some methods for the determination of the coefficients of the expansion.