Question

Let

Q:=-2×1+×2<=1;×1-×2<=1;-×1-×2<=-1
C:=-2×1+×2<=0;×1-×2<=0;-×1-×2<=0
and let P be the conve× hull of e×treme points of Q.
Show that the point ×=(1,1)^T can be written as ×=p+c, where p∈P and c∈C

Answer 1

To show that the point (1, 1) can be written as p + c, where p belongs to the convex hull P of Q's extreme points and c belongs to C, we first find the extreme points of Q and the convex hull P. The extreme points of Q form a triangle, and the convex hull P is the triangle itself. Since (1, 1) is inside the triangle, it belongs to P. Finally, we show that (1, 1) also satisfies the inequalities defining C, so it belongs to C as well.

Step-by-step explanation:

First, we need to find the extreme points of Q. To do this, we solve the equations of the inequalities in Q simultaneously:

-2x1 + x2 <= 1

x1 - x2 <= 1

-x1 - x2 <= -1

Simplifying these equations, we get:

x1 <= 1 - x2

x1 >= 1 + x2

x2 >= -1 - x1

If we plot these equations on a graph, we will find that the extreme points of Q form a triangle with vertices at (0, 1), (-2, -1), and (2, -3).

Next, we need to find the convex hull of these extreme points. The convex hull is the smallest convex set that contains all the extreme points. In this case, the convex hull is the triangle itself.

Now, let's prove that the point x = (1, 1) can be written as x = p + c, where p belongs to P (convex hull of Q's extreme points) and c belongs to C.

Since (1, 1) is inside the triangle formed by the extreme points of Q, it is also inside the convex hull P. Therefore, we can choose p = (1, 1).

Now, we need to find c. Since (1, 1) is inside the triangle formed by the extreme points of Q, it satisfies the inequalities defining Q. Therefore, it also satisfies the inequalities defining C. We can choose c = (0, 0).