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EE364a Homework 1 solutions
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<strong>EE364a</strong>, Winter 2007-08Prof. S. Boyd<strong>EE364a</strong> <strong>Homework</strong> 1 <strong>solutions</strong>2.1 Let C ⊆ R n be a convex set, with x 1 ,...,x k ∈ C, and let θ 1 ,...,θ k ∈ R satisfy θ i ≥ 0,θ 1 + · · ·+θ k = 1. Show that θ 1 x 1 + · · ·+θ k x k ∈ C. (The definition of convexity is thatthis holds for k = 2; you must show it for arbitrary k.) Hint. Use induction on k.Solution. This is readily shown by induction from the definition of convex set. Weillustrate the idea for k = 3, leaving the general case to the reader. Suppose thatx 1 ,x 2 ,x 3 ∈ C, and θ 1 + θ 2 + θ 3 = 1 with θ 1 ,θ 2 ,θ 3 ≥ 0. We will show that y =θ 1 x 1 + θ 2 x 2 + θ 3 x 3 ∈ C. At least one of the θ i is not equal to one; without loss ofgenerality we can assume that θ 1 ≠ 1. Then we can writey = θ 1 x 1 + (1 − θ 1 )(µ 2 x 2 + µ 3 x 3 )where µ 2 = θ 2 /(1 − θ 1 ) and µ 2 = θ 3 /(1 − θ 1 ). Note that µ 2 ,µ 3 ≥ 0 andµ 1 + µ 2 = θ 2 + θ 31 − θ 1= 1 − θ 11 − θ 1= 1.Since C is convex and x 2 ,x 3 ∈ C, we conclude that µ 2 x 2 + µ 3 x 3 ∈ C. Since this pointand x 1 are in C, y ∈ C.2.2 Show that a set is convex if and only if its intersection with any line is convex. Showthat a set is affine if and only if its intersection with any line is affine.Solution. We prove the first part. The intersection of two convex sets is convex.Therefore if S is a convex set, the intersection of S with a line is convex.Conversely, suppose the intersection of S with any line is convex. Take any two distinctpoints x 1 and x 2 ∈ S. The intersection of S with the line through x 1 and x 2 is convex.Therefore convex combinations of x 1 and x 2 belong to the intersection, hence also toS.2.5 What is the distance between two parallel hyperplanes {x ∈ R n | a T x = b 1 } and{x ∈ R n | a T x = b 2 }?Solution. The distance between the two hyperplanes is |b 1 − b 2 |/‖a‖ 2 . To see this,consider the construction in the figure below.1
- Page 3 and 4: (c) S = {x ∈ R n | x ≽ 0, x T y
- Page 5 and 6: Therefore we must have max i |x i |
- Page 7 and 8: (f) This set is convex. x + S 2 ⊆
- Page 9: p 1 + p 2 + · · · + p n−1prob(
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Ee364a Homework 1 Solutions
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EE364a Homework 1 solutions. 2.1 Let C ⊆ Rn be a convex set, with x1, . . . , xk ∈ C, and let θ1, . . . , θk ∈ R satisfy θi ≥ 0, θ1 + + θk = 1. Show that θ1x1 + + θkxk ∈ C. (The definition of convexity is that this holds for k = 2; you must show it for arbitrary k.) Hint. Use induction on k.
EE364a Homework 1 solutions. 2 Level sets of convex, concave, quasiconvex, and quasiconcave functions. Which of the following setsS are polyhedra? If possible, expressS in the formS ={x| Ax b, F x=g}. (a) S={y 1 a 1 +y 2 a 2 |− 1 ≤y 1 ≤ 1 , − 1 ≤y 2 ≤ 1 }, wherea 1 , a 2 ∈Rn. (b)S = {x ∈ Rn | x 0 , 1 Tx= 1,
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Solution. This is readily shown by induction from the definition of convex set. We illustrate the idea for k = 3, leaving the general case to the reader. Suppose that x 1 , x 2 , x 3 ∈ C, and θ 1 +θ 2 +θ 3 = 1 with θ 1 , θ 2 , θ 3 ≥ 0. We will show that y = θ 1 x 1 +θ 2 x 2 +θ 3 x 3 ∈C.
<strong>EE364a</strong>, Winter 2007-08Prof. S. Boyd<strong>EE364a</strong> <strong>Homework</strong> 1 <strong>solutions</strong>2.1 Let C ⊆ R n be a convex set, with x 1 ,...,x k ∈ C, and let θ 1 ,...,θ k ∈ R satisfy θ i ≥ 0,θ 1 + · · ·+θ k = 1. Show that θ 1 x 1 + · · ·+θ k x k ∈ C.
EE364a, Winter 2018-19 Prof. S. Boyd EE364a Homework 4 solutions 5.1 A simple example. Consider the optimization problem minimize x2 + 1 subject to (x − 2)(x − 4) ≤ 0, with variable x ∈ R. (a) Analysis of primal problem.
This document contains solutions to homework problems from an electrical engineering course. It addresses questions about convex sets, parallel hyperplanes, Voronoi descriptions of halfspaces, and determining whether certain sets are polyhedra.
final-sols.dvi. EE364a Convex Optimization I March 14–15 or March 15–16, 2008. Prof. S. Boyd. Final exam solutions. You may use any books, notes, or computer programs (e.g., Matlab, cvx), but you may not discuss the exam with anyone until March 18, after everyone has taken the exam.
Solution. This is readily shown by induction from the definition of convex set. We illustrate the idea for k = 3, leaving the general case to the reader. Suppose that x 1 , x 2 , x 3 ∈ C, and θ 1 + θ 2 + θ 3 = 1 with θ 1 , θ 2 , θ 3 ≥ 0. We will show that y = θ 1 x 1 + θ 2 x 2 + θ 3 x 3 ∈ C.
+(xn−1 −xn)(y1 +···+yn−1)+xn(y1 +···+yn). Solution. (a) The set Km+ is defined by n homogeneous linear inequalities, hence it is a closed (polyhedral) cone. The interior of Km+ is nonempty, because there are points that satisfy the in-equalities with strict inequality, for example, x = (n,n−1,n−2,...,1).