Farkas's Lemma

The Inequality $\left\langle{f_0, x}\right\rangle{}\leq 0$ follows from

$$\left\langle{f_1, x}\right\rangle{}\leq 0, \ldots, \left\langle{f_n, x}\right\rangle{}\leq 0$$

Iff there exist Nonnegative numbers $\lambda_1$, ..., $\lambda_n$ with

$$\sum_{k=1}^n \lambda_k f_k=f_0.$$

This Lemma is used in the proof of the Kuhn-Tucker Theorem.

See also Kuhn-Tucker Theorem, Lagrange Multiplier