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@jemoka / Jemoka Knowledge Base / raw/concept/kbhminimum_volume_elipsolid.md
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--- title: "minimum volume ellipsoid" source: https://www.jemoka.com/posts/kbhminimum_volume_elipsolid/ --- Lowner-John Ellipsoid minimum volume surrounding ellipsoid Consider a set of ellipsoid \(C\). Minimum volume ellipsoid \(\epsilon\) with \(C \subset \epsilon\). We can parameterize \(\epsilon\) as \(\epsilon = \qty {v \mid \norm{Av + b}_{2} \leq 1}\); where we assume \(A \in \mathcal{S}_{++}^{n}\). The volume is proportional to \(\text{det} A^{-1}\). Thus to find minimal-volume ellipsoid, solve: \begin{align} \min_{A,b}\quad & \log \text{det} A^{-1} \\ \textrm{s.t.} \quad & \text{sup}_{v \in C} \norm{A v + b}_{2} \leq 1 \end{align} OR, for finite sets: \begin{align} \min_{A,b}\quad & \log \text{det} A^{-1} \\ \textrm{s.t.} \quad & \norm{A x_{i} + b }_{2} \leq 1, i = 1 \dots m \end{align} Inside maximum volume inscribing ellipsoid Consider a set of ellipsoid \(C\). Minimum volume ellipsoid \(\epsilon\) with \(C \subset \epsilon\). We can parameterize \(\epsilon\) as \(\epsilon = \qty {Bu + d \mid \norm{u}_{2} \leq 1}\); where we assume \(B \in \mathcal{S}_{++}^{n}\). \begin{align} \max_{B,d}\quad & \log \text{det} B \\ \textrm{s.t.} \quad & \text{sup}_{\norm{u}_{2}} \leq 1, I_{C}\qty(Bu+d) \leq 0 \end{align} where \(I_{C} = 0\) when \(c \in C\), \(\infty\) otherwise.