Theorem for the Optimization of Linear Function over a Bounded Polyhedron

Question

In optimization theory, I often see people say that the minimum a linear function over a compact convex set is attainable at some extreme point of the feasible set. I have no problem with its proof, but I wonder if there is a name for such theorem or there is a more general statement for this result?

Answer 1

It is called the maximum principle:

In the field of convex optimization, there is an analogous statement which asserts that the maximum of a convex function on a compact convex set is attained on the boundary.

A maximizer of a convex function coincides with a minimizer of a concave function, and a linear function is concave. So the minimum of a linear function is attained on the boundary.