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In this paper, by using (h, ϕ)-generalized directional derivative and (h, ϕ)-generalized gradient, the authors directly derives the Karush- Kuhn-Tucker conditions by applying a corollary of Farkas lemma under the Mangasarian-Fromovitz constraint qualification.Furthermore, the boundedness of Lagrange multipliers is showed.

The Avriel-Ben-Tal algebraic operations approach for a short version proof of the Karush-Kuhn-Tucker optimality conditions