This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Solution of Homework 2 Problem (2.3.2): Solution: To show that S 1 is convex, we need to show that x + (1 ) y S 1 , given x, y S 1 and [0 , 1] . Let z = x + (1 ) y and we can see that z 1 = x 1 + (1 ) y 1 and z 2 = x 2 + (1 ) y 2 . First, since x 1 + x 2 1 and y 1 + y 2 1 , we have z 1 + z 2 = x 1 + (1 ) y 1 + x 2 + (1 ) y 2 = ( x 1 + x 2 ) + (1 )( y 1 + y 2 ) + (1 ) = 1 Second, since x 1 and y 1 , we have z 1 = x 1 + (1 ) y 1 Then z S 1 and thus S 1 is convex. We can similarly show that S 2 is convex. But S 1 S 2 is NOT convex. To show this, we need to give an example. For instance, x = (0 , 1) , y = (1 , 1) , and = 1 2 . We notice that x S 1 and y S 2 . But z = x + (1 ) y = ( 1 2 , 1) is not in S 1 S 2 . squaresolid Problem (2.3.3): Solution: Suppose x and y are in S . Let z = x + (1 ) y , and we have Az = A ( x + (1 ) y ) = Ax + (1 ) Ay b + (1 ) b = b 1 If you are not familiar with this matrix notation. You can show it sep arately. For each constraint, it can be written asarately....
View Full
Document
 Summer '10
 Brown

Click to edit the document details