Let $G_i \in \mathbb{R}^{n_i \times n_i}$ for all $i \in [1...m]$ and $x_i \in \mathbb{R}$ for all $i \in [1...m]$. Consider the semidefinite program $$\min_{G_i\succeq0,~x_i\geq 0, \forall i\in [1...m]}\sum_{i=1}^m\frac{1}{x_i}\,,$$ subject to $$\begin{cases}G_i \succeq x_i I_{n_i}~\forall i \in [1...m]\\ \begin{bmatrix}\text{blkdiag}(G_1,...,G_m)& H^T\\ H&\sum_{i=1}^{m}x_i\end{bmatrix}\succeq 0 \end{cases}$$ for some matrix $H \in \mathbb{R}^{1 \times \sum_{i=1}^mn_i}$. Can I equivalently state the objective function as
$$\max_{G_i\succeq0,~x_i\geq 0, \forall i\in [1...m]}\sum_{i=1}^mx_i?$$
The idea would be that minimizing the sum of the reciprocals seems equivalent to maximizing the sum of the non-reciprocals. However, I'm concerned about the constraints, which somewhat couple the $x_i$'s together. If it is not equivalent, what assumptions do I need on the constraints?
Here you can find a similar question for the simpler case where the $x_i$'s belong to separate sets: minimizing the sum of reciprocals is equivalent to maximizing the sum
No, but since you are working in the semidefinite programming world, you can just as well introduce new variables $z_i$ with $z_i \geq \frac{1}{x_i}$ which is SOCP-representable as it is equivalent to $\left|\left| \begin{matrix}x_i-y_i\\2\end{matrix} \right|\right| \leq x_i+y_i$, and then minimize $\sum z_i$ instead.