Identify the extreme points and extreme directions of $S = \{x : 4x_2 \geq x_1^2, x_1 +2x_2 + x_3 \leq 2\}$
Which I specifically need help finding the extreme directions.
The extreme points are defined by the intersection of the two defining constraints of $S$. With this in mind, I have found equations of $x_1$ and $x_2$ in terms of $x_3$ that represent the extreme points of $S$.
They are
$$x_1 = -1 \pm \sqrt{1-2x_3}$$
$$x_2 = \frac{3-x_3 \mp \sqrt{1-2x_3}}{2}$$
where $x_3 \leq \frac{1}{2}$
I just don't really have a clue on how to find the extreme directions from here.
These are the definitions I have from my professor:
Characterization of Extreme Points
Characterization of Extreme Directions
He usually doesn't give any examples, just usually the definitions and theorems for concepts.
So looking at these equations and sketching out $S$, definitely tells us that $S$ is bounded in the positive $x_3$ direction. Therefore if we have a extreme direction $d = (d_1, d_2, d_3)$, then it must be that $d_3 < 0$. Which I believe $d_3 = -1$ suffices.