# Maximize $\sum_{i=1}^n 1/x_i$ subject to an SDP constraint

I would like to solve the following problem: \begin{align}\max_{x_1, \ldots, x_n}&\quad\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}\\\text{s.t.}&\quad\sum_{i=1}^n x_i A_i \succeq A_0\\&\quad x_1, \ldots, x_n \geq 0\end{align}

Here $$A_0, A_1, \ldots, A_n$$ is a collection of symmetric matrices.

I'm not sure what is a good way to solve this, since the objective is the maximization (rather than minimization) of a convex function. Can this be done in polynomial time, and, if not, what are some good heuristics for this?