Consider the set $S=\{x:-x_1+2x_2\le 4,x_1-3x_2\le 3,x_1,x_2\ge 0\}$. Identify all
extreme points and extreme directions of $S$. Represent the point $(4,1)^t$ as a
convex combination of the extreme points plus a nonnegative combination of
the extreme directions.
Recall:
Definition: $d\in \mathbb R^n$ is a extreme direction of $S$ if $d=\lambda_1d_1+\lambda_2d_2,$ then $\lambda d_1=d_2.$, where $d_1,d_2$ are directions of $S.$ and $\lambda>0,\lambda_i>0$
This is what I've done:
I draw the region and I found 3 extreme points $(0,0),(0,2),(3,0).$
And I don't know how to identify the extreme directions, I don't even see it geometrically.
How can I see it geometrically?
For instance the extreme points are "the vertex of the figure". So the extreme directions are … ?
And finally, how do I prove it?
Best Answer
Check out the graph at https://www.desmos.com/calculator/sarf5jl1cq
Extreme directions are $(2,1)$ and $(3,1)$, which are the direction vectors emanating from the two nonzero extreme points in the boundary.