Differential Calculus in Nonlinear Programming

Using only well-known theorems of the differential calculus, we derive necessary conditions for a relative minimum of a nonlinear differentiable objective function of nonnegative variables constrained by nonlinear differentiable inequalities. The results are expressed entirely in terms of partial derivatives, which are subsequently identified with the Lagrange multipliers of the Kuhn-Tucker nonlinear programming theorem. Our conditions may be considered therefore as the Kuhn-Tucker theorem in differential rather than Lagrangian form.