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Given two ellipsoids $\mathcal{E}_1 = \{ X | X^T A_1 X + 2B_1^T X + C_1 \leq 0\}$ and $\mathcal{E}_2 = \{ X | X^T A_2 X + 2 B_2^T X + C_2 \leq 0\}$ both non-empty, it is possible to test if $\mathcal{E}_1 \subseteq \mathcal{E}_2$. Indeed, by the use of the so called S-procedure, $\mathcal{E}_1 \subseteq \mathcal{E}_2$ $\iff$ $\exists \lambda > 0$ such that $$ \begin{bmatrix} A_2 &B_2\\ B_2^T &C_2\end{bmatrix} \preceq \lambda \begin{bmatrix} A_1 &B_1 \\ B_1^T &C_1\end{bmatrix}$$

Question

Assume that $\mathcal{E}_1 \not\subseteq \mathcal{E}_2$. I want to find a point in $\mathcal{E}_1 \setminus \mathcal{E}_2$. How can I do that? I think one should follow the proof of the S-procedure (the necessity part) and eventually construct one on those lines ... Can someone help?

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