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You are given matrices $A$, $B$ and $C$. $C$ is symmetric and positive semidefinite. How would you go about constructing a matrix $X \succeq 0$ such that $X \succeq AXA^T$ and $C \succeq BXB^T$? The first can be satisfied by starting with any $X_0 \succeq 0$ and iterating $X_{t+1} = X_0 + AX_tA^T$, but I have no idea how to satisfy both.

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