For the sake of completeness, let me mention the direction which is opposite to the OP's Question.
There are combinatorial optimizations problems which deal with a finite (but large) input set at the start, and the point is to optimize a real function over the input. Such problems are often messy.
Then, some time ago, Hungarian mathematicians started to embed the input set into a Euclidean n-space, and they'd extend the said function to a linear or convex function over the convex hull of the input. Since the optimum over the whole hull is reached at a vertex then... etc.
We see that sometimes situations which are strictly finite (looking for an exact answer) get hm-reduced to infinite situations.