ECE 5734 - Assigment 1
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AOE5734 HW#1 Due January 30, 2015 1. (a) Show that if C 1 and C 2 are convex sets in R n , then so is their sum C 1 + C 2 , where C 1 + C 2 = {x 1 + x 2 | x 1 ∈ C 1 ,x 2 ∈ C 2 }. (b) Consider the affine function f (x)= Ax + b, where A ∈ R m×n , x ∈ R n , and b ∈ R m . Show that if C is a convex set, then so is f (C ) where f (C )= {f (x) | x ∈ C }. 2. Given a vector x =(x 1 ,x 2 ,...,x n ) ∈ R n , then the 1-norm, 2-norm (Euclidean norm), and ∞-norm are defined as: ‖x‖ 1 = |x 1 | + |x 2 | + ···|x n | ‖x‖ 2 =(|x 1 | 2 + |x 2 | 2 + ··· + |x n | 2 ) 1 2 =(x T x) 1 2 ‖x‖ ∞ = max i=1,...,n |x i |, respectively. Which of the following is a convex set? (a) {x ∈ R 2 | x 1 ≥ 0,x 2 ≥ 0, ‖x‖ 1 ≥ 1}; (b) {x ∈ R 2 | x 1 ≥ 0,x 2 ≥ 0, ‖x‖ 2 ≥ 1}; (c) {x ∈ R 2 | x 1 ≥ 0,x 2 ≥ 0, ‖x‖ ∞ ≥ 1}. 3. Solve problem 2.1 from the textbook. You may just illustrate the idea for the case k =3. 4. Solve problem 2.5 from the textbook.
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Convex optimization, Introduction
Transcript of ECE 5734 - Assigment 1
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AOE5734 HW#1 Due January 30, 2015
1. (a) Show that if C1 and C2 are convex sets in Rn, then so is their sum C1 + C2, where
C1 + C2 = {x1 + x2 | x1 C1, x2 C2}.
(b) Consider the affine function f(x) = Ax + b, where A Rmn, x Rn, and b Rm.Show that if C is a convex set, then so is f(C) where
f(C) = {f(x) | x C}.
2. Given a vector x = (x1, x2, . . . , xn) Rn, then the 1-norm, 2-norm (Euclidean norm), and-norm are defined as:
x1 = |x1|+ |x2|+ |xn|
x2 = (|x1|2 + |x2|
2 + + |xn|2)
1
2 = (xTx)1
2
x = maxi=1,...,n |xi|,
respectively. Which of the following is a convex set?
(a) {x R2 | x1 0, x2 0, x1 1};(b) {x R2 | x1 0, x2 0, x2 1};(c) {x R2 | x1 0, x2 0, x 1}.
3. Solve problem 2.1 from the textbook. You may just illustrate the idea for the case k = 3.4. Solve problem 2.5 from the textbook.
