Convex sets and convex functions taking the infinity value

Chapter 1 Convex sets and convex functions taking the infinity value
Chapter 1.
Convex sets and convex functions taking the infinity value
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Chapter 1 Convex sets and convex functions taking the infinity value
Convex set
Definition. A subset C of IR
n
is convex if
∀x, y ∈ C ∀t ∈ [0, 1] tx + (1 − t)y ∈ C
Proposition. If C is convex, then its interior int C and its closure
C
are convex
Convexity is preserved by the following operations :
Let I be an arbitrary set. If C
i
⊆ IR
n
, i ∈ I , are convex, then
C = ∩
i∈I
C
i
is convex
Let C and D be two convex sets in IR
n
and let a and b be two real
numbers. Then the following set is convex :
aC + bD := {ac + bd | c ∈ C , d ∈ D}
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Chapter 1 Convex sets and convex functions taking the infinity value

Illustration
Y
X
X
Y
convex non convex
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Chapter 1 Convex sets and convex functions taking the infinity value
Examples of convex sets
The following are some examples of convex sets :
(1) Hyperplane : S = {x|p
T
x = α}, where p is a nonzero vector in IR
n
,
called the normal to the hyperplane, and α is a scalar.
(2) Half-space : S = {x|p
T
x ≤ α}, where p is a nonzero vector in IR
n
,
and α is a scalar.
(3) Open half-space : S = {x|p
T
x < α}, where p is a nonzero vector in
IR
n
and α is a scalar.
(4) Polyhedral set : S = {x|Ax ≤ b}, where A is an m × n matrix, and b
is an m vector. (Here the inequality should be interpreted

elementwise.)
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Chapter 1 Convex sets and convex functions taking the infinity value
Examples of convex sets
(5) Polyhedral cone : S = {x|Ax ≤ 0}, where A is an m × n matrix.
(6) Cone spanned by a finite number of vectors :
S = {x|x =

m
j=1
λ
j
a
j
|λ
j
≥ 0, j = 1, . . . , m}, where a
1
, . . . , a
m
are
given vectors in IR
n
.
(7) Neighborhood : N
ε
(¯x) = {x ∈ IR
n
|x − ¯x < ε}, where ¯x is a fixed
vector in IR

n
and ε > 0.
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Chapter 1 Convex sets and convex functions taking the infinity value
Convex cone
Some of the geometric optimality conditions that we will study use convex
cones.
Definition. A nonempty set C in IR
n
is called a cone with vertex zero
if x ∈ C implies that αx ∈ C for all α ≥ 0. If, in addition, C is convex,
then C is called a convex cone.
0
0
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Convex cone

Nonconvex cone
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Chapter 1 Convex sets and convex functions taking the infinity value
Convex combination and convex hull of a set
Definition. x is said to be a convex combination of x
1
, . . . , x
m
if there
exist α
1
≥ 0, . . . , α
m
≥ 0 such that
x = α
1
x
1
+ · · · + α
m
x
m
, and α
1
+ · · · + α
m
= 1.
The convex hull of C (denoted conv C) is the intersection of all convex
subsets containing C.
Proposition (Carath´eodory’s lemma). Let C ⊆ IR

n
. Then each
element of conv C is a convex combination of at most n + 1 points of C
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Chapter 1 Convex sets and convex functions taking the infinity value
Illustration
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