I have an optimization problem with the following SDP constraint:
\begin{equation*} \begin{bmatrix}\gamma I-\Gamma & \gamma\Sigma\\ \gamma\Sigma & Z\end{bmatrix}\geq 0 \end{equation*} where $\Gamma\in\mathbb{S}^{n}, Z\in\mathbb{S}_+, \gamma\in\mathbb{R}_+$ are optimization variables, while $\Sigma\in\mathbb{S}^n$ is a constant vector.
How can I reformulate this constraint into a nonlinear (nonconvex) one? I have tried using Schur complement, but because of $Z$ I still get a positive semidefinite constraint.