First, I train a standard feed-forward neural network over a training set of data points. I get an approximate function, say $F(x)$, represented implicitly by that neural network.
1. How do I find its (global) maximum?
Note that the function $F(x)$ is not given explicitly; it is only implicitly realized by the neural network.
$F(x)$ should be differentiable, and gradient descent may be applied to it. But I'm not sure how this could be done given only the neural network.
2. Could the structure of the neural network be exploited so that I could find the maxima faster?
PS: the question is not the same as finding the maximum value of a set of data points -- I need the neural network for other purposes, so the neural network must be trained first, and the maximum be found afterwards.