Two approaches to handling uncertainty. Consider an LP:

\begin{align} &\min c^{T}x \\ &s.t. a_{i}^{x} \leq b_{i} \end{align}

what if our constraints are uncertain. Both of these reduce to an SOCP. See slides.

Deterministic Worst-Case

\begin{align} &\min c^{T}x \\ &s.t.\ a_{i}^{T} x \leq b_{i}, \forall a_{i} \in \epsilon_{i} \end{align}

Stochastic

\begin{align} &\min c^{T}x \\ &s.t.\ \text{prob}\qty(a_{i}^{T} x \leq b_{i}) \geq \eta, i = 1 \dots m \end{align}