Strong Duality: Without Simplex and without theorems of alternatives By

We provide an alternative proof of the strong duality theorem whose main step is a proposition which says that every canonical linear programming minimization problem whose image under its objective function of the set of feasible solutions is non-empty and bounded below has an optimal solution. Unlike earlier proofs, this proof neither uses the simplex method, nor does it use Farkas’s lemma. We also use this proposition to obtain an independent proof of Farkas’s lemma.