Revisiting the Hahn-Banach theorem and nonlinear infinite programming

The aim of this paper is to state a sharp version of the König supremum theorem, an equivalent reformulation of the Hahn-Banach theorem. We apply it to derive statements of the Lagrange multipliers, Karush-Kuhn-Tucker and Fritz John types, for nonlinear infinite programs. We also show that a weak concept of convexity coming from minimax theory, infsup-convexity, is the adequate one for this kind of results.