HW2 - Solution of Homework 2 Problem(2.3.2 Solution To show...

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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 αx + (1 − α ) y ∈ S 1 , given x, y ∈ S 1 and α ∈ [0 , 1] . Let z = αx + (1 − α ) y and we can see that z 1 = αx 1 + (1 − α ) y 1 and z 2 = αx 2 + (1 − α ) y 2 . First, since x 1 + x 2 ≤ 1 and y 1 + y 2 ≤ 1 , we have z 1 + z 2 = αx 1 + (1 − α ) y 1 + αx 2 + (1 − α ) y 2 = α ( x 1 + x 2 ) + (1 − α )( y 1 + y 2 ) ≤ α + (1 − α ) = 1 Second, since x 1 ≥ and y 1 ≥ , we have z 1 = αx 1 + (1 − α ) 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 α = 1 2 . We notice that x ∈ S 1 and y ∈ S 2 . But z = αx + (1 − α ) 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 = αx + (1 − α ) y , and we have Az = A ( αx + (1 − α ) y ) = αAx + (1 − α ) Ay ≤ αb + (1 − α ) 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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This note was uploaded on 01/11/2011 for the course MATH Math 164 taught by Professor Brown during the Summer '10 term at UCLA.

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HW2 - Solution of Homework 2 Problem(2.3.2 Solution To show...

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