ESI 6417
Convex and Polyhedral Sets
1
Ref. Chapter 2, BJS
•
Convex Sets:

A set
X
in
R
n
is called a “convex” set if, for any (or every) two points (vectors)
x
1
and
x
2
in
X
, any convex combination of
x
1
and
x
2
lies in
X
.
A convex combination =
λ
x
1
+ (1 –
λ
)
x
2
, where
λ
∈
[0, 1].
A convex combination of
x
1
and
x
2
represents a point on the line segment
joining
x
1
and
x
2
.
Which of the following sets are convex?
If
X
and
Y
are two convex sets, is
X
∩
Y
convex?
If
X
and
Y
are two convex sets, is
X
∪
Y
convex?

Extreme points of a convex set: A point
x
∈
X
is called an “extreme” point of
X
if
x
cannot be represented as a convex combination of two distinct points in
X
.
Identify extreme points of the following sets.
•
Hyperplanes and halfspaces

A set of the form {
x
∈
R
n
:
a
T
x
=
α
}, where
a
is a vector and
α
is a constant, is
called a “hyperplane.”
For example,
X
={
x
∈
R
2
: 2
x
1
+ 1
x
2
= 2}
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ESI 6417
Convex and Polyhedral Sets
2
Ref. Chapter 2, BJS
X
= {
x
∈
R
3
:
x
1
+ 2
x
2
+
x
3
= 6}
A set of the form {
x
∈
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 Spring '07
 SIRIPHONGLAWPHONGPANICH
 Optimization, Polytope, Convex combination, Polyhedral Sets Ref

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