Can one find the minima of a convex function efficiently? Say I have a real valued convex function $f$ on the hypercube $[-1,1]^n$ and let $f'$ be its restriction on the discrete hypercube $\{-1,1\}^n$.


Is there any $poly(n)$ algorithm that for any class of $f$ (and $f'$) can find the minima of it? 
Is there any relation known between the point where $f$ minimizes and the vertex where $f'$ minimizes?  Like if knowing one of them helps find the other in $poly(n)$ time?
Minimizing a submodular function can be done in polynomial time.  This applies to functions on the discrete hypercube $\{-1,1\}^n$.