This preview shows pages 1–4. Sign up to view the full content.
CHAPTER VIII
CONVEX SETS
We present some geometric aspects of linear programming.
Definition. A subset S of
ℜ
n
is said to be
convex, if given
x
and
y
in S, then the line segment [
x,y
] connecting
x
and
y
lies in S.
The
line segment
[
x,y
]
connecting
x
and
y
is the set of all points
[
x,y
] = {
λ
x +
(1 
λ
)
y
 0
≤
λ
≤
1}.
Picture of a Convex Set.
Example (1). Let
A
be an
m x n
matrix and let
b
be in
ℜ
n
. Then the set
S = {
x
∈
ℜ
n
 A
x
=
b
}
is a convex set. In fact, we have that
A
x
=
b
and
A
y = b
implies that
A(
λ
x
+ (1 
λ
)
y
) =
λ
A
x
+ (1 
λ
)A
y
= t
b
+ (1 
λ
)
b
=
b,
for
0
≤
λ
≤
1. Thus, [
x, y
]
⊂
S.
Example (2). Let
A
be an
m x n
matrix and let
b
be in
ℜ
n
. Then the set
S = {
x
∈
ℜ
n
 A
x
=
b, x
≥
0
}
is a convex set. As in Example (1), we have that
A
x
=
b
and
A
y = b
implies that
A(
λ
x
+ (1 
λ
)
y
) =
λ
A
x
+ (1 
λ
)A
y
= t
b
+ (1 
λ
)
b
=
b.
In addition we have that
λ
x
+ (1 
λ
)
y
≥
0,
for
0
≤
λ
≤
1. Thus, [
x, y
]
⊂
S.
Example (3).
Let
A
be an
m
×
n
matrix and let
b
be in
ℜ
n
.
Then the set
S = {
x
∈
ℜ
n
 A
x
≤
b
}
is a convex set since
A(
λ
x
+ (1

λ
)
y
) =
λ
A
x
+ (1 
λ
)A
y
≤
λ
b
+ (1 
λ
)
b
=
b,
for
0
≤
λ
≤
1.
Definition. A realvalued function f on a convex set S is said to be
convex if
f(
λ
x
+ (1 
λ
)
y
)
≤
λ
f(
x
) + (1 
λ
)f(
y
),
for all
x
and
y
in
S.
Graph of a Real Valued Convex Function.
Chapter VIII Convex Sets
page 1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document Example of a Convex
Function. Let c
∈
ℜ
n
and let f(
x
) =
c
⋅
x
(dot product).
Then f is a convex function. In fact, the function f is linear and
every linear function is convex.
Proposition.
Let f be a convex function on a convex set S; then every local minimum for f is a global minimum.
Proof.Let f(
x
o
) be a local minimum for f. By the definition of local minimum there is a d > 0 such that f(
x
o
)
≤
f(
x
)
for
every
x
in S with

x x
o
 <
δ
. Now, for arbitrary
y
in S, we need to show that
f(
x
o
)
≤
f(
y
) to show that
f(
x
o
)
is a global minimum. We use the convexity of S for this. In fact, we have that

x
o

(
λ
y
+ (1 
λ
)
)
x
o
)
≤

λ
 
x
o
 y

and thus that

x
o

(t
y
+ (1t)
x
o
) < d
whenever
0 <
λ
<
δ
/(
x
o
 + 
y
 + 1). Thus, we have that
f(
x
o
)
≤
f(
λ
y
+ (1 
λ
)
x
o
)
≤
λ
f(
y
) +(1 
λ
)f(
x
o
),
or equivalently,
tf(
x
o
)
≤
tf(
y
)
whenever
0 < t < d/(
x
o
 + 
y
 + 1). Thus, f(
x
o
)
≤
f(
y
).
Q.E.D.
The Proposition shows that every local minimum in a Linear Programming Problem is a global minimum.
Definition. Let S be a convex set in
ℜ
n
; then a point
x
in S is said to be an
extreme point of S if given
y
and
z
in S
with
x
∈
[
y, z
], then
x = y = z.
Note that
x
is an extreme point of S if it is not properly inside any line segment in S.
Examples of Extreme Points.
Chapter VIII Convex Sets
page 2
The extreme points of the Triangle
The extreme points of the circle
are the vertices of the triangle.
is the boundary of the circle
.
Definition. A set of the form
H = {
x
∈
ℜ
n

c
⋅
x
= a}
is said to be a
hyperplane in
ℜ
n
. Here
c
is a nonzero vector in
ℜ
n
and
c
⟩
x
represents the dot product of
c
and
x
.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview. Sign up
to
access the rest of the document.
This document was uploaded on 11/03/2009.
 Spring '09

Click to edit the document details