Nonlinear duality and multiplier theorems

The main purpose of this paper is to extend the John theorem on nonlinear programming with inequality contraints and the Mangasarian-Fromovitz theorem on nonlinear programming with mixed constraints to any real normed linear space. In addition, for the John theorem assuming Frechet differentiability, the standard conclusion that the multiplier vector is not zero is sharpened to the nonvanishing of the subvector of those components corresponding to the constraints which are not linear affine. The only tools used are generalizations of the duality theorem of linear programming, and hence of the Farkas lemma, to the case of a primal real linear space of any dimension with no topological restrictions. It is shown that these generalizations are direct consequence of the ordiry duality theorem of linear programming in finite dimension.