Suppose I have the equation $A \leq P U X U^\dagger P$, where A is positive semidefinite, U is unitary, P is a projection, and X is an unknown positive semidefinite matrix, and $\leq$ is the Loewner order.
Now suppose that $X = \Sigma_i I \otimes \ldots X_i \ldots \otimes I$, where each $X_i$ is an unknown positive semidefinite matrix, all the $X_i$ have the same dimensions, but each $X_i$ may be in a different position in the tensor product (or they may be in the same position). For example, $X = X_1 \otimes I \otimes I + I \otimes X_2 \otimes I + I \otimes X_3 \otimes I$.
Is there an analytic way to find a feasible solution for all the $X_i$? Or are there good tools to calculate this? I have been using CVX and other optimization/constraint solvers but, in my specific case, they tend to violate the constraint (the first equation) too frequently.