ISyE 7682
Fall 2010 – 3rd Homework
Problem 0.1
For a set
S
⊆ ℜ
n
, show the following equivalences:
a)
S
is closed if, and only if,
I
S
is a lower semicontinuous function;
b)
S
is convex if, and only if,
I
S
∈
Conv(
ℜ
n
)
;
c)
S
is a nonempty closed convex set if, and only if,
I
S
∈
C
onv (
ℜ
n
)
.
Solution
. a) Since epi
I
S
=
S
×
[0
,
∞
), it follows that
S
is closed if, and only if, epi
I
S
is closed,
and the latter condition is equivalent to the lower semicontinuity of
I
S
, in view of Proposition
2.3.3.
b)
S
is closed if, and only if, epi
I
S
=
S
×
[0
,
∞
) is convex, and the latter condition is equivalent
to the convexity of
I
S
, in view of Proposition 2.1.6.
c) In view of a) and b),
S
is a closed convex set if, and only if,
I
S
is a lower semicontinuous
convex function.
Moreover, the nonemptyness of
S
is equivalent to the properness of
I
S
.
Since
C
onv (
ℜ
n
) is exactly the set of all proper lower semicontinuous convex functions, c) follows.
Problem 0.2
Show the following statements:
a) for given functions
f
1
, f
2
:
ℜ
n
→
¯
ℜ
, we have
f
1
≤
f
2
if, and only if,
epi
f
1
⊇
epi
f
2
; in
particular,
f
1
=
f
2
if, and only if,
epi
f
1
= epi
f
2
;
b) for given function
f
:
ℜ
n
→
¯
ℜ
and a family of functions
f
λ
:
ℜ
n
→
¯
ℜ
,
λ
∈
Λ
, we have
f
= sup
λ
∈
Λ
f
λ
if, and only if,
epi
f
=
∩
λ
∈
Λ
epi
f
λ
.
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 Fall '08
 Staff
 Mathematical analysis, Convex set, Qd, Convex function, Epigraph, cl f2

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