## 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:

\begin{equation} \min_{x \in X} \max_{z \in Z} f(x,z) \end{equation}

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:

\begin{equation} \min_{x \in X} \mathbb{E}_{z \sim P} [f(x,z)] = \min_{x \in X}\int_{Z} f(x,z) p(z) \dd{z} \end{equation}

problem additive noise

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

\begin{equation} f(x,z) = f(X) + z \end{equation}

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

This means that:

\begin{equation} \min_{x \in X} \mathbb{E}_{z \sim P} [f(x,z)] = \min_{x \in X} \qty(\mathbb{E}_{z \sim P} [f(x)] + \mathbb{E}_{z \sim P}[z]) = \min_{x \in X} \qty(f(x) + 0) \end{equation}

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} \qty[\qty(f(x,z) - \mathbb{E}_{z \in Z}\qty[f(x,z)])^{2}] \\ &= \int_{z \in Z} f(x,z)^{2}p(z) \dd{z} - \mathbb{E}_{z \in Z} \qty[f(x,z)]^{2} \end{align}

If you have a covariance matrix and a mean vector, you can formulate:

\begin{equation} \min_{x} x^{\top} u + \lambda x^{\top} \Sigma x \end{equation}

#### feasible set approaches

statistical feasibility

“the probability that a design point is feasible”

\begin{equation} P((x,z) \in \mathcal{F}) = \int_{z} ((x,z) \in \mathcal{F}) p(z) \dd{z} \end{equation}

value at risk

best objective value which can be guaranteed with probability \(\alpha\) given the error distribution.

where as conditional value at risk CVaR is expected value of top \(1-\alpha\) quartile of the distribution