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The ellipsoid you need to find can be numerically computed by solving the following optimization problem.

Let me denote $X_K\triangleq\{x|H_uKx\leq h_u\}=\{x|a_{u,i}^\top x\leq b_{u,i}, i = 1,...,n_u\}$ the polyhedral set that satisfies the condition $x\in X_K,Kx\in U$. Then, your problem can be regarded as a well-known problem of finding a maximum volume inscribed ellipsoid (See more detail here at Section 8.4.2 in this book of Stephen Boyd together with the theoretical proof employing the hint you have). More specifically, you need to find the ellipsoid $E=\{x|x^\top P x\leq \alpha\}$ so that it is inscribed in $X_K$. To align our problem with the representation in the book, the ellipsoid $E$ can be rewritten as the following form: $$ E=\{x=\sqrt\alpha Bv, \forall v^\top v \leq 1,v\in\mathbb R^n\} $$ where $B$ can be computed as $B=((U\sqrt S V^\top)^\top )^{-1}$ with[U,S,V] come from the singular value decomposition of $P$, $(P=USV^\top)$. The transformation between these two representations can be found in more detail in this paper. Then, with $\beta=\sqrt\alpha$, $\beta$ can be found by solving the following convex optimization problem: $$ \beta^*=\underset{\beta}{\arg \max \log \det(\beta B)} $$ $$ \text{subject to: } \|\beta B a_{u,i}\|_2\leq b_{u,i}, \,i = 1,...,n_u. $$ This is known to be a convex optimization problem and, particularly for your question, it has only one scalar variable $\beta$. With this, you can have the value of $\alpha=\beta^2$ for your problem. Implementation of this problem was already discussed here with CVX solver.

Tinne0411
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