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Let $B$, $C$, $W$ and $V$ be given (known) matrices with $V$ and $W$ being semidefinite positive. We would like to determine the matrices $X$, $Z$ and $T$ by solving the following inequality \begin{equation} \begin{bmatrix}Z+\begin{bmatrix}X^{T}VX & XC\\ C^{T}X & C^{T}V^{-1}C \end{bmatrix} & & \begin{bmatrix}X & 0\\ C^{T}V^{-1} & B \end{bmatrix}\\ \\ \begin{bmatrix}X^{T} & V^{-1}C\\ 0 & B^{T} \end{bmatrix} & & T-Z+\begin{bmatrix}V^{-1} & 0\\ 0 & W^{-1} \end{bmatrix} \end{bmatrix}\succeq0. \end{equation}

$Z$ and $T$ are also semidefinite positive. All the terms are linear in the unknown matrices except $X^{T}VX$. It would have been very trivial to solve this inequality if there was no $X^{T}VX$ in it. Does anyone know how to transform this inequality into linear matrix inequality (LMI) please? Or does anyone know how to solve this inequality in order to determine the matrices $X$, $Z$ and $T$ please? Thanks.

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