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Department of Mathematical
Sciences
University of Delaware
Prof. T. Angell
November 10, 2010
Mathematics 530
Test Problems
Exercise 1
: Determine the extreme points of the set
S
, where
S
is the set of all solutions
to
x
1
+
x
2
≤
4
2
x
1
+
x
2
≤
6
x
2
≤
3
x
1
,x
2
≥
0
and write the point (9
/
7
,
1
/
2) as a convex combination of these extreme points.
Exercise 2
: Let
A
be and
m
×
n
matrix,
b
∈
R
m
, and
K
⊂
R
n
a convex set. Prove or
disprove that the set
{
x
∈
R
n

A
x
≤
b
} ∩
K
is a convex subset in
R
n
.
Exercise 3
: Suppose that
V
is a real vector space and that
f
:
V
→
R
satisﬁes the
inequality
f
((1

λ
)
x
+
λy
)
≤
(1

λ
)
f
(
x
) +
λf
(
y
)
,
for all
x,y
∈
V,
0
≤
λ
≤
1
.
Show that the set of all minimizers for
f
is a convex set. (Be careful: what happens when
f
is not bounded below?)
Exercise 4
: Let
K
⊂
R
n
be a convex set and
f
:
K
→
R
. Then
f
is called
quasi
convex
provided that, for all
λ
∈
(0
,
1)
, f
( (1

λ
)
x
+
λ
y
)
≤
max
{
f
(
x
)
, f
(
y
)
}
for all
x
,
y
∈
K
.
(a) Show that every function satisfying the inequality in the previous problem is quasi
convex.
(b) Show that every strictly increasing function of a single real variable is quasiconvex
and ﬁnd an example that shows that it need
not
be convex.
(c) Show that the level sets of
f
,
{
x
∈
K
:
f
(
x
)
≤
b,b
∈
R
}
, are convex if and only if
the function
f
is quasiconvex.
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View Full DocumentExercise 5
:
(a) Sketch the convex polyhedron generated by the following set of points
{
(1
,
0)
,
(3
,
2)
,
(4
,
3)
,
(

1
,
2)
,
(

3
,

2)
}
.
(b) Find the equation of a supporting hyperplane to this polyhedron at the point (3
,
2).
Is this the only supporting hyperplane at that point?
Exercise 6
: Let
A
be a symmetric
n
×
n
matrix. The matrix
A
is said to be
positive
deﬁnite
provided
x
>
A
x
>
0 for all
x
∈
R
n
,
x
6
= 0. Show that the mapping
{
x
,
y
} 7→
y
>
A
x
deﬁnes and inner product on
R
n
provided that
A
is positive deﬁnite.
HINT:
the
new inner product
y
*
x
:=
y
>
A
x
.
Exercise 7
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