Given a N real valued variables and a set of linear inequality constraints, I would like to find a minimal bounding box which encapsulates the convex polytope defined by these constraints.
I think (but am not certain) I can frame this as 2*N linear programming problems, where the constraints remain the same for each one, but the cost function is just the identity function on each variable, and I attempt to both maximise and minimise this to find the upper and lower bounds of each dimension of the box. I could then apply any common algorithm such as the simplex algorithm.
This seems like it may be overkill however; is there a simpler way to solve this problem?
A more natural approach is to enumerate the vertices of the polytope. Here is a link to an algorithmic approach: http://cgm.cs.mcgill.ca/~avis/doc/avis/AF92b.pdf