I have the set $S ⊂ R^6$ $$ \begin{align*} −&x_1 +x_2 +x_3 = 2\\ &x_1 +x_2 −x_4 = 1\\ &x_1 −6x_2 +x_5 = 3\\ −&2x_1 +x_2 +x_6 = 2\\ x_1 \geq 0, x_2 \geq 0, &x_3 \geq 0, x_4 \geq 0, x_5 \geq 0, x_6 \geq 0 \end{align*} $$
How do I know if $S$ is bounded? If it isn't bounded, how do I find two recession directions of $S$?
Let's denote your set $S = \{ x : Ax \leq b\}$. You want to find $a \in \mathbb{R}^6$ and $b \in \mathbb{R}^6$ such that $A(a+tb) \leq b$ for all $t \in \mathbb{R}_+$. This simplifies to finding $a$ and $b$ such that $Aa \leq b$ and $Ab\leq 0$. You can find such $a$ and $b$ with the simplex method.
