hw4sol_fa09 - UC Berkeley Department of Electrical...

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Unformatted text preview: 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 k 2 k x- y k 2 } , i.e., an intersection of half-spaces. (Note that k x- x k 2 k x- y k 2 x T x- 2 x T x + x T x x T x- 2 x T y + y T y 2( y- x ) T x + x T x- y T y so the set y S { x |k x- x 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 = { } , 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 k x- b k 2 } = { x |k x- a k 2 2 k x- b k 2 2 } = { x | (1- 2 ) x T x- 2( a- 2 b ) T x + ( a T a- 2 b T b ) } If = 1, this is a half-space. If= 1, this is a half-space....
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This note was uploaded on 09/03/2011 for the course EE 227A taught by Professor Staff during the Spring '11 term at University of California, Berkeley.

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hw4sol_fa09 - UC Berkeley Department of Electrical...

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