Farkas Lemma (III)
Let \(A\in\mathbb{R}^{m\times d}\), \(z\in\mathbb{R}^{m}\), \(a_0\in(\mathbb{R}^{d})^*\), and \(z_0\in\mathbb{R}\).
Then \(a_0x\leq z_0\) is valid for all \(x\in\mathbb{R}^d\) with \(Ax\leq z\), if and only if
Then \(a_0x\leq z_0\) is valid for all \(x\in\mathbb{R}^d\) with \(Ax\leq z\), if and only if
- there exists a row vector \(c\geq0\) such that \(cA=a_0\) and \(cz\leq z_0\) or
- there exists a row vector \(c\geq0\) such that \(cA=0\) and \(cz<0\).