Any technique for linearization, or linear approximation?

Question

Consider the following Matrix constraint: $$ \begin{bmatrix} -U+\psi\Sigma_b^{-1} & V \\ V^T & -V^TU^{-1}V+\tau_2 -\psi \end{bmatrix} \leq 0 $$ where $\Sigma_b$ is a known positive definite matrix (shape matrix of saturation bound). $U \geq 0 \in \mathbb{S}_{+}^{ n \times n} $ and $V \in \mathbb{R}^n$ and $\tau_2,\psi >0 \in \mathbb{R}$ are the decision variables. I can propose the following linearization technique for the problem. $$ \begin{bmatrix} 0 & 0 \\ 0 & -V^TU^{-1}V \end{bmatrix} \leq \begin{bmatrix} U-\psi\Sigma_b^{-1} & -V \\ -V^T & -\tau_2 +\psi \end{bmatrix} $$ as the left hand side is surely negative semi definite we can replace the original nonlinear constraint with the following linear constraint. $$ \begin{bmatrix} U-\psi\Sigma_b^{-1} & -V \\ -V^T & -\tau_2 +\psi \end{bmatrix} \geq 0 $$ But this linearization has conservatism which is generated from ignoring the features of term $V^TU^{-1}V$ and its impacts on the problem. So I wanted to ask do you guys have any suggestion for improving this linearization process?