Do LP problems always have basic feasible solutions

Question

Assume that the feasible set is not empty.
My textbook says if a basic solution satisfy nonnegative condition, then it is a basic feasible solution. I am wondering whether there are some theorems that can assure a LPP has at least one basic feasible solution as long as the feasible set is not empty.
Thanks!

Answer 1

That is not true.

Consider $\min_{x,y} y$ subject to $x \ge 0$.

It has no BFS at all though it is feasible.

Edit:

• We know that every non-empty polyhedron in standard form has at least one BFS.