I have a function $F$ which is defined as follows: $$ F(x) = \sum_{i=1}^N f(z_i^T x) $$ where ${z_i}$ are known $m \times 1$ vectors, $x$ is an $m \times 1$ vector, and for $t\in \mathbb{R}$, $f(t) = \frac{1}{1+e^{-t/\sigma}}$ with a known $\sigma$ (a sigmoid function).

For a local or global maximum of $F(x)$, I would like to characterize the basin of attraction---the set of all starting points from which a gradient ascent algorithm converges to the local maximum.

I know that the gradient of $F$ is: $$ \nabla F = \frac{1}{\sigma} \sum_i \frac{e^{-z_i^Tx/\sigma}}{(1+e^{-z_i^Tx/\sigma})^2}z_i^T $$ and the Hessian can also be easily derived.

How can I use the gradient and Hessian to come up with a simple condition for the basin of attraction of $x^*$, e.g., if $\|x-x^*\|^2<g$ for some $g$, then $x$ is in the basin of attraction of $x^*$?