Lagrange Multipliers Revisited

The present paper was inspired by the work of Kuhn and Tucker [1]. These authors transformed a certain class of constrained maximum problems into equivalent saddle value (minimax) problems. Their work seems to hinge on the consideration of still a third type of problem. A very simple but illustrative form of this problem is the following: let \( x\;\epsilon \) positive orthant of some finite dimensional Euclidean space, and let f and g be real valued functions of x with the property that whenever \( f \geq 0, \)then also \( g \geq 0; \) under what conditions can one then conclude that 3 a non-negative constant u such that uf \( \leq g; \; for \; \underline{all} \;x \geq 0 ?\)