Separation oracle

Given a convex set $\mathcal{K}$, separation oracle $S_\mathcal{K}$ returns: $$S_\mathcal{K}=\begin{cases} \emptyset & \textrm{if }\mathbf{y}\in \mathcal{K}\\ \textrm{separating hyperplane } \mathcal{H} & \textrm{if }\mathbf{y}\notin \mathcal{K} \end{cases}$$ Let $\mathcal{H} = \{\mathbf{x} : \mathbf{a}^\intercal \mathbf{x} = c\}$.

used in ellipsoid method

References:

1. https://www.chrismusco.com/amlds2023/notes/lecture09.html
2. https://en.wikipedia.org/wiki/Hyperplane_separation_theorem