Suppose you have a parallelogram and a point. It's easy to tell which of the parallelogram's vertices is closest to the point (Euclidean distance) by checking the distance for every vertex - but this gets very exhaustive for higher dimensions because the number of vertices of an n-dimensional parallelotope is $2^{dim}$.
I've tried a) following the steepest descent of the Euclidean norm w.r.t. the given point and b) iteratively going from one vertex its neighbor vertex with the smallest Euclidean norm w.r.t. the given point, but I found cases where this doesn't work for one of the methods.
So - how do I solve this problem on a computer for "high" dimensions, e.g. $>30$?