13 Concavity and Inequalityconstrained Optima
Proposition 13.1
Let
U
⊂
R
n
be open and convex. Let
f
:
U
→
R
and
h
i
:
U
→
R
be concave
C
1
functions,
i
= 1
,
···
,l
. Suppose also that there exists
x
∈
U
such that
h
i
(
x
)
>
0 for all
i
= 1
,
···
,l
. — Slater’s condition
Then,
x
*
is a
global
maximum of
f
in the set
D
=
U
∩ {
x
∈
R
n
:
h
i
(
x
)
≥
0
,
∀
i
= 1
,
···
,l
}
if and only
if there exists
λ
*
= (
λ
*
1
,
···
,λ
*
l
)
∈
R
l
such that
(KT1)
Df
(
x
*
) +
l
X
i
=1
λ
*
i
Dh
i
(
x
*
) = 0
n
(KT2)
λ
*
i
≥
0 for all
i
= 1
,
···
,l
and
l
X
i
=1
λ
*
i
h
i
(
x
*
) = 0
14 Quasiconcavity and Inequalityconstrained Optima
Proposition 14.1
Let
f
:
D
→
R
be a
C
1
function where
D
⊂
R
n
is convex and open.
Then,
f
is quasiconcave if and only if for every
x,y
∈
D
,
f
(
y
)
≥
f
(
x
) =
⇒
Df
(
x
)(
y

x
)
≥
0
.
Moreover, if
f
is quasiconcave, then for all
x
∈
D
with
Df
(
x
)
6
= 0 and for all
y
∈
D
,
f
(
y
)
> f
(
x
) =
⇒
Df
(
x
)(
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 Spring '08
 Hayashi
 Derivative, Necessary and sufficient condition, Convex function, Slater

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