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S is the polyhedral set

$ S = \{ \mathbf{x} \in \mathbb{R}^{n} ; \mathbf{Ax}=\mathbf{b}, \mathbf{x} \ge \mathbf{0} \} $

and

$ H : \mathbf{c}^{T}\mathbf{x} = \beta $

with

$ \min_S ( \mathbf{c}^{T}\mathbf{x} )= \beta $

My textbook states that given the above, the set $S \cap H$ contains no line since $\mathbf{x} \ge \mathbf{0}$ in $S$. But I don't understand why.

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1 Answer 1

up vote 3 down vote accepted

Any line in $\mathbb R^2$ can be written as $$L = \left\{ \left(\begin{matrix}x \\ y\end{matrix}\right) + t \left(\begin{matrix}u \\ v\end{matrix}\right) : t \in \mathbb R \right\}$$ where $u$ and $v$ are not both zero. For some choice of $t$, either $x+tu < 0$ or $y + tv < 0$. So $L$ contains a point $\mathbf x$ that does not satisfy $\mathbf x \geq 0$. So $L$ cannot be contained in $S$ (and hence not in $S\cap H$, no matter what $H$ is).

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I just realised that I put R^2 where it should have been R^n. But I guess what you are saying is true for R^n as well. Thank you. –  evading Nov 24 '12 at 21:09
    
Yes, it works just the same in $\mathbb R^n$. –  marlu Nov 25 '12 at 13:05

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