I want a real solution of the inequality \begin{equation} X P X^t \le Q \end{equation} where $P$ and $Q$ are given positive semidefinite Hermitian matrices (may not be of same dimension). The solution, $X$, has to be a real matrix of appropriate dimension such that the dimensions in left and right hand sides match. Also it is nice if we can show optimality of such solutions
Is there a general method for solving such inequalities? Advanced thanks for any help, suggestion, references, and etc..
Suppose both $P$ and $Q$ are positive semidefinite.
If $Q\succeq YPY^T$ for some real $Y$, then $(I-QQ^+)YP=0$, i.e. $\left[P\otimes(I-QQ^+)\right]\operatorname{vec}(Y)=0$. Hence $$ \pmatrix{\Re\left[P\otimes(I-QQ^+)\right]\\ \Im \left[P\otimes(I-QQ^+)\right]} \operatorname{vec}(Y)=0.\tag{1} $$ Equation $(1)$ suggests that there usually isn't any non-trivial solution.
Since $Q$ is positive definite on its column space, given any real solution $Y$ to $(1)$, $X:=rY$ will be a real solution to $Q\succeq XPX^T$ when $r>0$ is sufficiently small. The usual argument shows that the maximum feasible value of $r$ is $\dfrac1{\sqrt{\rho(Q^+YPY^T)}}$.