# How get the extreme directions of an unbounded feasible region

The following constraints form a feasible region.

$$-x_1+x_2 \le 2$$

$$-x_1+2x_2 \le 6$$

$$x_1,x_2 \ge 0$$

The feasible region have three extreme points: $$e_1=\left[\begin{array}{cc} 0\\ 0 \end{array}\right]$$ $$e_2=\left[\begin{array}{cc} 0\\ 2 \end{array}\right]$$ $$e_3=\left[\begin{array}{cc} 2\\ 4 \end{array}\right]$$

What is the procedure that I need to follow to extract the extreme direction from this data?

Extrene Direction: An extreme direction of a convex set is a direction of the set that cannot be represented as a positive combination of two distinct directions of the set.

• the costraints equations with the term on the right 0 and with "=" instead of $\le$. – tommycautero Jan 23 at 14:07
• What do you mean by "extreme directions" in this context? – callculus Jan 23 at 18:38
• Check the edit that I've done – user637533 Jan 23 at 19:03

Draw the LP using the graphical method (Just plot the conditions as line on the $$x_1$$ and $$x_2$$ axis) let $$x_1$$ be the horizontal axis and $$x_2$$ be the vertical axis. Find the line that remains in the feasible region as $$x_1$$ tends to infinity. and take the ratio of increase in $$x_2$$ and $$x_1$$. So if you have a line $$-x_1 +2x_2 \geq 8$$ that stays in the feasible region as $$x_1$$ tends to infinity, then the extreme direction is $$(2,1)$$ as when $$x_1$$ increases by $$2$$ then $$x_2$$ increases by $$1$$. Another example: you have a line $$x_2 \geq 2$$,that stays in the feasible region as $$x_1$$ tends to infinity, then the extreme direction is $$(1,0)$$.