Sign up ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Suppose that $A$ is an $m\times n$ matrix, $D$ is a $p\times n$ matrix, $b$ is an $m$-vector, and $d$ is a $p$-vector. Prove that there does not exist $n$-vector $x$ satisfying

$$Ax \geq b, Dx \leq d$$

if and only if there exist $m$-vector $y$ and $p$-vector $w$ satisfying

$$ y \le 0, w \ge 0,$$

$$ (A^T)y + (D^T)w = 0,$$

$$ (b^T)y + (d^T)w < 0.$$

share|cite|improve this question
If this is a homework, please tag it as so. Where did you get stuck? – aelguindy Mar 6 '12 at 14:09
I do not know how to start in the first place. – newbowl Mar 6 '12 at 14:43

1 Answer 1

up vote 1 down vote accepted

Note that $Ax \geq b$, $Dx \leq d$ is equivalent to $$\left(\begin{matrix}-A \\D \end{matrix}\right)x \leq \left( \begin{matrix}-b\\d\end{matrix}\right).$$ Farkas' Lemma tells you that this does not have a solution if and only if there is a vector $u \geq 0$ that satisfies $$u^T \left(\begin{matrix}-A \\D \end{matrix}\right) = 0 \quad \text{and} \quad u^T \left( \begin{matrix}-b\\d\end{matrix}\right) < 0.$$ Then write $u = \left( \matrix{ -y \\ w} \right)$.

share|cite|improve this answer
I've never heard of Farkas Lemma. Thanks for shedding some light on what area I should look up on.:) – newbowl Mar 6 '12 at 15:47

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.