Explain the Kuhn-Tucker Theorem in economics.

Kuhn-Tucker Theorem:

Assume that x solves the inequality constrained optimization problem and also satisfies the constrained qualification situation. After that, there are a set of Kuhn-Tucker multipliers as λ_i > 0, i = 1, . . . , k , that is as:

Moreover, we have the complementary slackness situations as:

λ_i > 0 for each i - 1,2,...,k

λ_i = 0 when g_i(x) < D_i.

By comparing the Kuhn-Tucker theorem to the Lagrange multipliers within the equality constrained optimization difficulty, we notice that the main difference is that the signs of the Kuhn-Tucker multipliers are non-negative whereas the signs of the Lagrange multipliers may be anything. Its additional information can occasionally be very helpful. The Kuhn-Tucker theorem merely gives an essential condition for a maximum.