optimization uncertainty irreducible uncertainty: uncertainty inherent to a system epistemic uncertainty: subjective lack of knowledge about a system from our standpoint uncertainty can be presented as a vector of random variables, z, where the designer has no control. Feasibility of a design point, then, depends on (x, z) \in \mathcal{F}, where \mathcal{F} is the feasible set of design points. set-based uncertainty set-based uncertainty treats uncertainty z as belonging to some set \bold{Z}. Which means that we typically use minimax to solnve:

we don’t assume anything about the distribution of z. probabilistic uncertainty uncertainty expected value optimization Instead of z \in Z blindly, we assume some underlying distribution of z. The most natural way to do this is to compute the expectation directly:

problem additive noise For a moment, let’s assume that the noise is added directly:

Also, let’s consider z \sim \mathcal{N}(0, \Sigma). This means that:

meaning, in this specific case, optimizing for expected value is bad. uncertainty variance optimization \begin{align} \Var[f(x,z)] &= \mathbb{E}{z \in Z} \left[\left(f(x,z) - \mathbb{E}{z \in Z}\left[f(x,z)\right]\right)^{2}\right] \ &= \int_{z \in Z} f(x,z)^{2}p(z) \dd{z} - \mathbb{E}_{z \in Z} \left[f(x,z)\right]^{2} \end{align} If you have a covariance matrix and a mean vector, you can formulate:

feasible set approaches statistical feasibility “the probability that a design point is feasible”