  [REVISED VERSION] The aim of this paper is to state a sharp version of the
K\"onig 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 type, 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.
