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Nonlinear duality and multiplier theorems. (English) Zbl 0656.90098

The main purpose of this paper is to extend the John theorem on nonlinear programming with inequality constraints and the Mangasarian-Fromovitz theorem on nonlinear programming with mixed constraints to any real normed linear space. In addition, for the John theorem assuming Fréchet 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 a direct consequence of the ordinary duality theorem of linear programming in finite dimension.

MSC:

90C48 Programming in abstract spaces
49N15 Duality theory (optimization)
46B10 Duality and reflexivity in normed linear and Banach spaces
49J50 Fréchet and Gateaux differentiability in optimization
90C30 Nonlinear programming
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