Integrals can sometimes be computed through their saddle points. Are there examples of converse, when saddle points are found via integrals? Under some reasonable assumptions integrals with large exponents can often be computed via saddle point approximations, e.g.
$$\int e^{-\lambda f(x)}\approx e^{-\lambda f(x_0)},\qquad \lambda\to\infty$$
where I assume that $x_0$ is a unique minimum of the function $f(x)$. These types of computations are omnipresent in physics, for example. I am interested if there are examples in the reverse direction, when finding the minimum of $f(x)$ directly is challenging, but computing something like
$$\frac{\int e^{-\lambda f(x)}x}{\int e^{-\lambda f(x)}}\approx x_0,\qquad \lambda\to\infty$$
is tractable and allows to locate the minimum. Perhaps generalization to many variables or functional integrals would be more appropriate.
I hope the question is meaningful despite the lack of precision and rigor. Any comments are welcome!
 A: This is not an answer, but continuing the discussion in the comment box is a bit cumbersome. The point I want to make is to see if this would work for a simple test case, $f(t)=ix\sin\pi t$, with saddle point at $t=1/2$. So the question is whether we can "reverse engineer" this value of $t$ from integrals of the form $\int e^{ix\sin\pi t}dt$. The OP suggest to look at the ratio
$$I(x)=\frac{\int_{-1}^1 e^{ix\sin\pi t}t\,dt}{i\int_{-1}^1 e^{ix\sin\pi t}\,dt}=\frac{1}{J_0(|x|)}\int_{0}^1 \sin(x\sin\pi t)\,t\,dt.$$
Can we somehow recover the value $t=1/2$ from the large-$x$ behaviour of $I(x)$?
From the plot of $I(x)$ it seems that this function does not converge to a definite limit as $x\rightarrow\infty$.

A: This is a very particular instance of what's known as moment problem, or sometimes "inverse moment problem". One has an unknown measure $\mu$ in some space of measures, and one tries to reconstruct $\mu$ from knowlegde of $\mu_k:=\int f_k(x) d\mu(x)$, $k=0,1,\dots$, for a well-behaved sequence of functions $f_k$.
This readily generalises to several dimensions.
