I want to solve the following problem where $y \in \mathbb{R}^{m \times 1}$ and $Z \in \mathbb{R}^{m \times m}$. Also, $Z \succ 0$ and $X \succeq 0$ \begin{equation} \min_{X \in \mathbb{R}^{m \times m}} y^\top(Z + X)^{-1} y \end{equation}
It can be reformulated as an SDP \begin{equation*} \begin{aligned} \min_{t, X \succeq 0}&\ t \\ \text{st:}& \left[\begin{array}{cc} t & y^\top \\ y & X + Z \end{array}\right] \succeq 0 \end{aligned} \end{equation*}
Is it possible to solve this problem without formulating it as an SDP ? Either algorithmically or something simpler than SDP.