Convex Optimization Homework #1,
1. (30%) Is the set {a ∈ R
k
| p(0) = 1; |p(t)| ≤ 1 for α ≤ t ≤ β}, where p(t) = a1 + a2t + · · · + akt
k−1
, convex?
2. (30%) Prove or disprove that the function f : Rn+1 → R defined as f(x, t) = − log(t
2 − x
T x), with
dom f = {(x, t) ∈ Rn × R | ||x|| < t}, is convex.
3. (40%)
(a) (20%) Show that f : R2 → R, f(x) = 1
x1 −
1
x2
, with dom f =
n
x ∈ R2
| x2 > 0, x1 −
1
x2
> 0
o
is a
convex function.
(b) (20%) Prove or disprove that f : R4 → R,
f(x) = 1
x1 −
1
x2− 1
x3− 1
x4
,
with dom f =
(
x ∈ R2
x4 > 0, x3 −
1
x4
> 0, x2 −
1
x3− 1
x4
> 0, x1 −
1
x2− 1
x3− 1
x4
> 0
)
, is convex.
Guidelines of Homework Submission:
• You are allowed to discuss with other students, ask for hints from the TAs. But you have to write your
answers and argument solely on your own, without looking at any part of anyone else’s answers. Sharing
your written (or typed) answers with others is strongly prohibited. Both parties will get a zero-score
penalty for this mis-conduct.
• Submit your answer online as a document file (in *.pdf only) that contains all answers in this problem set.
• Submit your files online onto the Ceiba system. No paper shall be handed in. You can write (sketch) your
answers on a sheet first and convert the image(s) to a single pdf file.
• Late submissions will be processed according to the following rules.
(1) Homework received by 9pm, April 22 (t1) will be counted in full.
(2) Homework received after 0am, April 23 (t2) will not be counted.
(3) Homework received between t1 and t2 will be counted with a discount rate
t2 − t
t2 − t1
where t is the submission time. Note that t2 − t1 is three hours.
