# KKT conditions for $\max \log \det(X)$ with LMI constraints

I am trying to derive the KKT conditions for the following convex optimization problem where $$A$$ is a given matrix:

$$\begin{array}{ll} \underset{X,Y,Z}{\text{minimize}} & - \log \det \left(I + Z + A X A^T + Y A^T + A Y^T \right)\\ \text{subject to} & \begin{pmatrix} Z & Y\\\\ Y^T &X\end{pmatrix}\succeq 0\\\\ & \text{Tr}(Z)\le 10\\\\ & \begin{pmatrix}I-X&YA^T\\ AY^T&I + Z + A X A^T + Y A^T + AY^T\end{pmatrix}\succeq0\end{array}$$

The $$\log\det(\cdot)$$ is a scalar so we can compute its derivative with respect to any of the matrix variables, but my main challenge is how to treat the LMIs. Will be happy for any help/reference to start with.

I shall start writing my try: \begin{align} \frac{\partial f_0}{\partial X} &= - R^{-1} AA^T\\ \frac{\partial f_0}{\partial Y} &= - R^{-1} (A^T + A)\\ \frac{\partial f_0}{\partial Z} &= - R^{-1}, \end{align} where $$R = I + Z + A X A^T + Y A^T + A Y^T$$ and $$f _0$$ denotes the objective function.

Next, we should write the constraints in a standard form $$F_i(x) = \sum_i x_i F_i\succeq0$$ where $$F_i\succeq0$$.