A new proof of the Kuhn–Tucker and Farkas theorems

For the minimization problem for a differentiable function on a set defined by inequality constraints, a simple proof of the Kuhn–Tucker theorem in the Fritz John form is presented. Only an elementary property of the projection of a point onto a convex closed set is used. The approach proposed by the authors is applied to prove Farkas’ theorem. All results are extended to the case of Banach spaces.