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Unformatted text preview: 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 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 precedesequal b bardbl x y bardbl 2 2 , i.e. , the square of the Euclidean distance of x to the polyhedron defined by Ay precedesequal b . You may assume that the inequalities Ay precedesequal b are strictly feasible. (f) f ( x ) = sup Ay precedesequal b y T x . (You can assume that the polyhedron defined by Ay precedesequal b is bounded.) Solution. (a) Find k { 1 , . . . , m } for which f ( x ) = a T k x + b k . Then g = a k is a subgradient at x ....
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 Fall '09
 StephenBoyd

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