I'm trying to prove the following statement
Show that the set ${\{(x,w) \in \mathbb R^n\times \mathbb R^m \mid Ax \leq0, c^T x >0,w^TA=c, w\geq0 \}}$ is empty, where $A\in \mathbb R^{m\times n}$ and $ c \in \mathbb R^n$ are given.
Asked
Active
Viewed 257 times
1
-
At a glance you are expected to analyze the sign of $w^TAx$. – hardmath May 08 '16 at 17:50
-
1shouldn't it be $(x,w)\in \mathbb{R}^{n+m}$? – user190080 May 08 '16 at 18:35
-
@hardmath, do you know how to answer this? In the simplex algorithm, when might some a variable leave a basis? – BCLC May 20 '16 at 19:25
-
1@BCLC: I can take a look. – hardmath May 20 '16 at 19:28
-
@hardmath thanks ^-^ – BCLC May 20 '16 at 22:57
1 Answers
3
Suppose that it is non-empty and let $(x,w) \in \Bbb R^n \times \Bbb R^m$ satisfy the property. Then (where for a vector $v$, $(v)_i$ denotes the $i$-th component):
$$w^T A x= \sum_{i=1}^m (Ax)_i (w)_i$$
but $(Ax)_i \ge 0$ and $(w)_i \le 0$ for all $i$, so $w^TAx \le 0$.
On the other hand, $w^T Ax = cx > 0$.