MIT6_079F09_hw03

# MIT6_079F09_hw03 - 6.079/6.975, Fall 200910 S. Boyd & P....

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6.079/6.975, Fall 2009–10 S. Boyd & P. Parrilo Homework 3 additional problems 1. Reverse Jensen inequality. Suppose f is convex, λ 1 > 0, λ i 0, i = 2 , . . . , k , and λ 1 + ··· + λ n = 1, and let x 1 , . . . , x n dom f . Show that the inequality f ( λ 1 x 1 + ··· + λ n x n ) λ 1 f ( x 1 ) + ··· + λ n f ( x n ) always holds. Hints. Draw a picture for the n = 2 case ±rst. For the general case, express x 1 as a convex combination of λ 1 x 1 + ··· + λ n x n and x 2 , . . . , x n , and use Jensen’s inequality. 2. Reformulating constraints in cvx . Each of the following cvx code fragments describes a convex constraint on the scalar variables x , y , and z , but violates the cvx rule set, and so is invalid. Brieﬂy explain why each fragment is invalid. Then, rewrite each one in an equivalent form that conforms to the cvx rule set. In your reformulations, you can use linear equality and inequality constraints, and inequalities constructed using cvx functions. You can also introduce additional variables, or use LMIs. Be sure to

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## This note was uploaded on 01/31/2012 for the course ECE 202 taught by Professor Boyde during the Fall '06 term at Goldsmiths.

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MIT6_079F09_hw03 - 6.079/6.975, Fall 200910 S. Boyd & P....

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