CO 355 Mathematical Optimization (Fall 2010)
Assignment 4
Due:
Thursday, November 11th, in class.
Policy.
No collaboration is allowed. You may use the course notes / textbook and the lecture slides, but
please be very specific
when using citing results found there. (Don’t just say “from some claim in class
we know
....
”.) Every other resource that you might stumble upon must be properly referenced. You are
welcome to seek help from the current instructor and TAs for CO 355.
Question 1:
(10 points)
[Exercise 3.2.3]
Let
S
⊆
R
n
be a convex set and
f
:
S
→
R
be a convex function. For any points
x
1
, ..., x
m
∈
S
and scalars
λ
1
, ..., λ
m
≥
0 with
∑
m
i
=1
λ
i
= 1, prove that
f
m
X
i
=1
λ
i
x
i
!
≤
m
X
i
=1
λ
i
f
(
x
i
)
.
Question 2:
(15 points)
[Exercise 3.2.7]
(a):
Let
R
++
=
{
x
∈
R
:
x >
0
}
. Let
g
:
R
++
→
R
be defined by
g
(
x
) =

log(
x
). (This is the
natural logarithm.) Prove that
g
is convex.
(b):
Prove that log
y
≤
y

1 for all
y >
0.
(c):
For any scalars
x
1
, ..., x
m
>
0 and scalars
λ
1
, ..., λ
m
≥
0 with
∑
m
i
=1
λ
i
= 1, prove that
m
X
i
=1
λ
i
x
i
≥
m
Y
i
=1
(
x
i
)
λ
i
.
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 Winter '10
 Harvey
 Optimization, optimal value

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