Hw2_Sol - Math464 - HW 2 Solutions Due on Thursday, Jan 28...

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Math464 - HW 2 Solutions Due on Thursday, Jan 28 1 Linear Optimization (Spring 2010) Brief solutions to Homework 2 1. We first prove that f is convex when each f i is convex. (a) Since the f i ’s are convex, λ [0 , 1], and for x , y R n , we have f i ( λ x + (1 - λ ) y ) λf i ( x ) + (1 - λ ) f i ( y ) , i = 1 ,...,m. (1) For f ( x ) = m i =1 f i ( x ), we get f ( λ x + (1 - λ ) y ) = m X i =1 f i ( λ x + (1 - λ ) y ) m X i =1 λf i ( x ) + (1 - λ ) f i ( y ) , (by (1)) = λ m X i =1 f i ( x ) ! + (1 - λ ) m X i =1 f i ( y ) ! = λf ( x ) + (1 - λ ) f ( y ) , which gives the definition of convexity of f . (b) Now consider the case where each f i is PL convex, i.e., f i ( x ) = max j =1 ,...,m i ( c T ij x + d ij ) , i = 1 ,...,m. Hence, we get f ( x ) = m X i =1 f i ( x ) = m X i =1 max j i =1 ,...,m i ( c T ij i x + d ij i ) = max j 1 =1 ,...,m 1 ,...,j m =1 ,...,m m X i =1 ,...,m ± ( c T ij 1 x + d ij 1 ) + ··· + ( c T ij m x + d ij m ) ² = max j 1 =1 ,...,m 1 ,...,j m =1 ,...,m m m X i =1 c ij 1 + ··· + c ij m ! T
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Hw2_Sol - Math464 - HW 2 Solutions Due on Thursday, Jan 28...

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