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 $c^Tx> 0$.