HW2 - Solution of Homework 2 Problem (2.3.2): Solution: To...

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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....
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HW2 - Solution of Homework 2 Problem (2.3.2): Solution: To...

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