Does max { $w^Tx$ subject to $x$ is a point on a given polyhedron } optimize at an extreme point? Is it necessary that the linear program max { $w^Tx$ subject to : $x$ is a point on a given polyhedron } attain its maximum at an extreme point of the polyhedron for any arbitrary w ? Let $c$ = max $w^Tx$. The idea is that $w^Tx$ = c is a line in the hyperspace that touches the polyhedron at an extreme point

Yes. If you relax the constraint to be on _or in_ the polyhedron it becomes a convex optimization problem, moreover the solution is always on the boundary, which means that it's on the polyhedron.

Some special cases are where an entire face of the polyhedron optimizes the function. There's also a degenerate case ($w=0$) where any point optimizes the function.

The Wikipedia article on linear programming briefly mentions this. It's one of the properties that the Simplex algorithm for solving linear programs relies on.