# Farkas lemma query

I had a query related to Farkas's lemma. As i understand as per the lemma the following two statements are equivalent:

For a matrix $$A \in \mathbb{R}^{m \times n}$$,and vector $$c \in \mathbb{R}^{n}$$ the following two claims are equivalent:

i. The implication $$Ax\le 0 \implies c^Tx\le0$$ holds true

ii. There exists $$y \in \mathbb{R}^m_+$$ such that $$A^Ty=c$$

However, I find a counterexample as follows:

let $$A= \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}, \ c^T = [1 \ \ 2], \text{and } x = [-5 \ \ 0]^T.$$ Then $$A^Ty=c \implies y = \begin{bmatrix} 1.5 \\ -0.5\end{bmatrix}$$ which violates ii although i is satisfied. Can someone help point what I am missing here?

The condition $$i$$ can be restated as: for every $$x$$, if $$Ax\leq 0$$, then $$c^Tx\leq 0$$.
However, note that if $$x=[-3,2]^T$$, for the given values of $$A$$ and $$c$$, $$Ax=[-1, -5]^T \leq 0$$, and $$c^Tx=1>0$$. Therefore, the condition $$i$$ is not satisfied, since there exists $$x$$ such that $$Ax\leq 0$$ and $$c^Tx> 0$$.
• The statement of the lemma is that given a "pair" $(A, c)$ of a matrix $A$ and a vector $c$, then the condition $i$ is satisfied for every $x$ if and only if there exists a $y$ satisfying the condition $ii$. The pair $(A, c)$ that you are giving does not satisfy the first condition, as I showed in the answer. Commented Mar 7 at 16:22