Elementary proof of classical necessary optimality conditions for a mathematical programming problem with inequality constraints

A study to describe an elementary proof of classical necessary optimality conditions, the Kuhn-Tucker theorem for a constrained optimization problem in a finite-dimensional or Banach space, was reported. The study was carried out without using the 'convexity, separability, and duality properties', the idea of passage to the limit, the Bolzano-Weierstrass theorems, theorems of the alternative, implicit function theorems, Lyusternik's theorem, and other classical facts traditional for optimization. The Kronecker-Capelli theorem for inequality constraint problems was used in the study. It was concluded that a similar elementary proof of the Kuhn-Tucker theorem holds under the Mangasarian-Fromovitz constraint qualification.