I am trying to solve the next problem \begin{aligned} & \underset{P}{\text{maximize}} & & \log \det P \\ & \text{subject to} & & A^T P^{-1} + P^{-1} A \preceq 0 \\ &&& P^{-1} \succeq 0 \\ &&& \ldots \end{aligned}
Is this problem convex or can be transfrom to convex?
I try to introduce the new variable Q
\begin{aligned} & \underset{P}{\text{maximize}} & & \log \det P \\ & \text{subject to} & & A^T Q + P Q \preceq 0 \\ &&& Q \succeq 0 \\ &&& P Q = I &&& \ldots \end{aligned} but last constrainst is BMI.
EDIT: Removed my incorrect answer.
However, it is trivial/ill-posed due to the homogenous form you use. If there exist a feasible solution ($A$ stable) you can let $P$ tend to infinity and obtain an arbitrarily good solution)