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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.

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

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  • $\begingroup$ Thank you very for your suggestion Omnomnomnom. It would have been so great for this case if we had - (minus) at the beginning of your formula. We would have used then Schur complement and it would solve the problem definitively. I will continue thinking about it, thanks. $\endgroup$ – G. Trav Jun 17 '17 at 18:36

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