How can we solve the following 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 $X$ and $Y$ by solving the following inequality \begin{equation} \begin{bmatrix}\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} & & \begin{bmatrix}YVY^{T} & 0\\ 0 & W \end{bmatrix}^{-1} \end{bmatrix}\succeq0. \end{equation}

Does anyone know how to transform this inequality into linear matrix inequality (LMI) using Schur complements or other methods please? Or does anyone know how to solve this inequality in order to determine the matrices $X$ and $Y$ please? Thanks.

Answer 1

Note that if $P$ is positive semidefinite, then so is $A^TPA$ for any matrix $A$.

By rewriting our matrix as the product $$ \pmatrix{\pmatrix{X & YC\\0&A}\\ & I}^T \pmatrix{\pmatrix{YVY^T\\ & W}^{-1} \\ & \pmatrix{YVY^T\\ & W}^{-1}} \pmatrix{\pmatrix{X & YC\\0&A}\\ & I} $$ We can see that your matrix will be positive semidefinite whenever $V$ and $W$ are positive semidefinite.