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hw4sol_fa09

# hw4sol_fa09 - UC Berkeley Department of Electrical...

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UC Berkeley Department of Electrical Engineering and Computer Science EECS 227A Nonlinear and Convex Optimization Problem Set 4 Fall 2009 Issued: Tuesday, October 6 Due: Thursday, October 15, 2009 Reading: Boyd and Vandenberghe: Chapter 2, § 4.3, 4.4 Problem 4.1 (d) This set is convex because it can be expressed as \ y S { x |k x - x 0 k 2 ≤ k x - y k 2 } , i.e., an intersection of half-spaces. (Note that k x - x 0 k 2 ≤ k x - y k 2 x T x - 2 x T x 0 + x T 0 x 0 x T x - 2 x T y + y T y 2( y - x 0 ) T x + x T 0 x 0 - y T y 0 so the set y S { x |k x - x 0 k 2 ≤ k x - y k 2 } is a half-space) (e) In general this set is not convex, for example with S = {- 1 , 1 } and T = { 0 } , we have { x | dist ( x, S ) dist ( x, T ) } = { x R | x ≤ - 1 / 2 or x 1 / 2 } which clearly is not convex. (g) This set is convex, in fact a ball { x |k x - a k 2 θ k x - b k 2 } = { x |k x - a k 2 2 θ k x - b k 2 2 } = { x | (1 - θ 2 ) x T x - 2( a - θ 2 b ) T x + ( a T a - θ 2 b T b ) 0 } If θ = 1, this is a half-space. If θ < 1, it is a ball { x | ( x - x 0 ) T ( x - x 0 ) R 2 } where the center x 0 and radius R given by x 0 = a - θ 2 b 1 - θ 2 , R = θ 2 k b k 2 2 - k a k 2 2 1 - θ

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