hw1 - Ay p b are strictly feasible. (f) f ( x ) = sup Ay p...

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EE364b Prof. S. Boyd EE364b Homework 1 1. For each of the following convex functions, explain how to calculate a subgradient at a given x . (a) f ( x ) = max i =1 ,...,m ( a T i x + b i ). (b) f ( x ) = max i =1 ,...,m | a T i x + b i | . (c) f ( x ) = sup 0 t 1 p ( t ), where p ( t ) = x 1 + x 2 t + ··· + x n t n 1 . (d) f ( x ) = x [1] + ··· + x [ k ] , where x [ i ] denotes the i th largest element of the vector x . (e) f ( x ) = inf Ay p b b x y b 2 2 , i.e. , the square of the Euclidean distance of x to the polyhedron de±ned by Ay p b . You may assume that the inequalities
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Unformatted text preview: Ay p b are strictly feasible. (f) f ( x ) = sup Ay p b y T x . (You can assume that the polyhedron de±ned by Ay p b is bounded.) 2. A convex function that is not subdiFerentiable. Verify that the following function f : R → R is convex, but not subdi²erentiable at x = 0: f ( x ) = b 1 x = 0 x > , with dom f = R + . 1...
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This note was uploaded on 04/09/2010 for the course EE 360B taught by Professor Stephenboyd during the Fall '09 term at Stanford.

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