Lattice Search

Suppose f(x,y) is a convex function. Now we want to find optimal integer x* and y* to minimize f(x,y) over integer set.
In continuou case, if we know x*(y), then we can just minimize f(x*(y),y) by taking derivative.
In discrete case, if we know x*(y), then can we say that -f(x*(y),y) is unimodular?