If the analytical form global minimum value of a nonconvex $C^{\infty} f:R^n\to R$ is already known, will it be easier to find its global minimizer $x^*\in R^n$?
If the global minimum value of a nonconvex $C^{\infty} f: R^n\to R$ is known,can it be easier to find the global minimizer?
optimization
Within a branch and bound scheme, having a good lower bound on the optimal value can be very helpful in speeding up the algorithm.  If your function is Lipschitz continuous with a known Lipschitz constant, and the region of interest can be bounded then you can readily apply branch and bound to the problem.  More generally, the problem is hopeless, since it's possible to hide the global minimum in a $C^{\infty}$ bump that could be located anywhere in $R^{n}$ and which could have arbitrarily small support.  