gradients and objectives

[gusl]

Given a differentiable objective function f: Rⁿ --> R, it's straightforward to numerically compute a gradient term, which can be iteratively followed to reach (local) maxima (as long as the size of the steps shrink exponentially as they approach the maximum).

But given a way of computing an update term, can you work back to an analytical description of f? I think Lyapunov function may be the answer.