Let $G \in \mathbb{R}^{n \times n}$, $H \in \mathbb{R}^{p \times n}$ and $x_i \in \mathbb{R}$ for all $i \in [1...m]$ be decision variables. Let $A\in \mathbb{R}^{p \times p}$ be a known invertible matrix. Consider the problem $$\min_{G\succeq 0,~x_i\geq 0, \forall i \in [1...m]} \sum_{i=1}^m \frac{1}{x_i}$$ $$\begin{bmatrix}G& H^T\\ H&A\left(\sum_{i=1}^{m}\frac{1}{x_i}\right)^{-1}\end{bmatrix}\succeq 0.$$
Can this be equivalently reformulated as a SDP?
A first step can be to add the SOCP constraint $$|| \begin{bmatrix}x_i-z_i&2\end{bmatrix}||_2\leq x_i+z_i\,,$$ so that $z_i \geq \frac{1}{x_i}$ and the problem becomes
$$\min_{G\succeq 0,~x_i\geq 0,~z_i,~\forall i \in [1...m]} \sum_{i=1}^m z_i$$ $$\begin{bmatrix}G& H^T\\ H&A\left(\sum_{i=1}^{m}z_i\right)^{-1}\end{bmatrix}\succeq 0.$$ $$|| \begin{bmatrix}x_i-z_i&2\end{bmatrix}||_2\leq x_i+z_i\,, \forall i \in [1...m]\,.$$
Is it possible to remove the nonlinearity in $z_i$ in the positive-semidefinite matrix?
Let $f(x)$ denote the harmonic mean. The objective is a scaled inverse of the harmonic mean, hence you are effectively maximizing the harmonic mean (which is concave and SOCP representable). Similarily, the term in the constraint is also the harmonic mean divided by $m$. An equivalent hypograph model is
$$\max t$$ $$\begin{bmatrix}G& H^T\\ H&A\frac{t}{m}\end{bmatrix}\succeq 0, f(x) \geq t$$