432hwk4 - . Give an example of a convex function g : R R...

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Math 432/532 Homework 15 points Due February 17, 2012 1. Which of the following functions are convex on the speciﬁed convex sets? Which are strictly convex? (a) f ( x 1 , x 2 ) = 5 x 1 2 + x 2 2 + 2 x 1 x 2 - x 1 + 2 x 2 + 3 on D = R 2 . (b) f ( x 1 , x 2 ) = x 1 2 / 2 + 3 x 2 2 / 2 + 3 x 1 x 2 on D = R 2 . (c) f ( x 1 , x 2 ) = ( x 1 + 2 x 2 + 1) 8 - ln[( x 1 x 2 ) 2 ] on D = { ( x 1 , x 2 ) | x 1 > 1 , x 2 > 1 } . (d) f ( x 1 , x 2 ) = 4 e 3 x 1 - x 2 + 5 e x 1 2 + x 2 2 on D = R 2 . (e) f ( x 1 , x 2 ) = c 1 x 1 + c 2 /x 1 + c 3 x 2 + c 4 /x 2 on D = { ( x 1 , x 2 ) | x 1 > 0 , x 2 > 0 } , where each c i is a positive real number. (f) f ( x 1 , x 2 , x 3 ) = x 1 2 + 3 x 2 - x 3 on D = R 3 . 2. Let C R n be convex, and suppose f : C R and g : C R are convex. Deﬁne h : C R by h ( x ) = max { f ( x ) , g ( x ) } . (a) Prove that epi h = epi f epi g . (b) Use part (a) to show that h is convex. 3. Let g : R n R be convex with g ( x ) 3 for all x R n . Show that the function f deﬁned by f ( x ) = ( g ( x ) - 3) 2 is convex on R n
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Unformatted text preview: . Give an example of a convex function g : R R such that ( g ( x )-3) 2 is not convex. 4. Let f : R n R be strictly convex, and suppose f ( x ) = f ( y ) = 0 for two points x and y in R n with x 6 = y . Show that there exists z R n such that f ( z ) &lt; . 5. (532 only) Let f : R n R have continuous rst-order partial derivatives at all points in R n . Show that f is convex if and only if ( f ( x )- f ( y )) ( x-y ) 0 for all x, y in R n ....
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