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LECTURE 2
LECTURE OUTLINE
•
Convex sets and Functions
•
Epigraphs
•
Closed convex functions
•
Recognizing convex functions
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View Full Document SOME MATH CONVENTIONS
•
All of our work is done in
<
n
: space of
n
tuples
x
=(
x
1
,...,x
n
)
•
All vectors are assumed column vectors
•
“
0
” denotes transpose, so we use
x
0
to denote a
row vector
•
x
0
y
is the inner product
∑
n
i
=1
x
i
y
i
of vectors
x
and
y
•k
x
k
=
√
x
0
x
is the (Euclidean) norm of
x
.W
e
use this norm almost exclusively
•
See Section 1.1 of the book for an overview of
the linear algebra and real analysis background
that we will use
CONVEX SETS
Convex Sets
Nonconvex Sets
x
y
α
x + (1 
α
)y,
0 <
α
< 1
x
x
y
y
x
y
•
A subset
C
of
<
n
is called
convex
if
αx
+(1
−
α
)
y
∈
C,
∀
x, y
∈
∀
α
∈
[0
,
1]
•
Operations that preserve convexity
−
Intersection, scalar multiplication, vector sum,
closure, interior, linear transformations
•
Cones: Sets
C
such that
λx
∈
C
for all
λ>
0
and
x
∈
C
(not always convex or closed)
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View Full Document CONVEX FUNCTIONS
α
f(x) + (1 
α
)f(y)
xy
C
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This note was uploaded on 06/16/2010 for the course MS&E 211 taught by Professor Yinyuye during the Fall '07 term at Stanford.
 Fall '07
 YINYUYE

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