Given $A\in\mathbb{R}^{n\times n}$ \begin{array}{ll} \underset{X\in\mathbb{R}^{n\times n}}{\text{min}} & \mathrm{tr}(AX^{-1}A'X).\\ \text{s.t.} & X\succ0\end{array}
("$X\succ0$" means that $X$ is symmetric positive definite matrix)
Special case when $A=\begin{bmatrix} a_1 & & \\ & a_2 & \\ &&a_2 \end{bmatrix}$ was discussed here: Find $\mathrm{tr}(AX^{-1}A'X).$
Is it possible to linearize/convexify such problem? Or is there any good approximation methods?