Duality in Semi-Infinite Programs and Some Works of Haar and Carathéodory

By constructing a new infinite dimensional space for which the extreme point—linear independence and opposite sign theorems of Charnes and Cooper continue to hold, and, building on a little-known work of Haar (herein presented), an extended dual theorem comparable in precision and exhaustiveness to the finite space theorem is developed. Building further on this a dual theorem is developed for arbitrary convex programs with convex constraints which subsumes in principle all characterizations of optimality or duality in convex programming. No differentiability or constraint qualifications are involved, and the theorem lends itself to new computational procedures.