For teaching purposes, I believe it can be good to use very light-weight proofs of deep results, as it often answers the question "why is it true" better than other types of proofs. By "light-weight proofs" I mean proofs that rely as little as possible on other deep results as possible.
Regarding proofs of strong duality in linear programming, I have seen many proofs. To name a few
- Strong duality as consequence of weak duality + simplex algorithm produces optimal solutions (Introduction to linear optimization, Bertsimas & Tsitsiklis)
- Strong duality as a consequence of weak duality + a projection argument (Large scale linear and integer optimization, Kipp Martin)
- Strong duality as a consequence of weak duality + Farkas' lemma (Theory of linear and integer programming, Schrijver)
- Strong duality as a corollary of strong duality of Lagrangian duality (Noonlinear programming, Bazaraa, Sheral, Shetty)
It can be debated how light-weight these proofs are, as most of them rely on some rather significant results. Thus, my question is
"What good light-weight proofs of strong duality in linear programming do you know?"