.1.2 Some Definitions

The feasible region of this LP is in general, a convex polyhedron. Visualize it as a polygon in 2 dimensions, for simplicity. Now, maximizing yT b is the same as maximizing the projection of the vector y in the direction represented by vector b. For whichever direction b we choose, the point y that maximizes yT b cannot lie strictly in the interior of the feasible region. The reason is that, from an interior point, we can move further in any direction, and still be feasible. In particular, by moving along b, we can get to a point with a larger projection along b. This intuition suggests that the optimal solution of an LP will never lie in the interior of the feasible region, but only on the boundaries. In fact, we can say more. We can show that for any LP, the optimal solutions are always at the “corners” of the feasible region polyhedron. This notion is formalized in the next subsection.