How can we solve this matrix inequality?

Question

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

$Z$ and $T$ are also semidefinite positive. It seems like we cannot use Schur complement to transform this inequality matrix into LMI (Linear Matrix Inequality). 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 $X$, $Y$, $Z$ and $T$ please? Thanks.

Answer 1

(Too big to fit in a comment:) It might be useful to note that we can write this as $$ \pmatrix{\pmatrix{X & YC\\0&A}\\ & I}^T \pmatrix{\pmatrix{YVY^T\\ & W}^{-1} \\ & \pmatrix{YVY^T\\ & W}^{-1}} \pmatrix{\pmatrix{X & YC\\0&A}\\ & I} \\ + \pmatrix{Z\\ & T - Z} $$