Atsushi Inoue
ECG 765: Mathematical Methods for Economics
Fall 2009
Lecture Notes 14
SecondOrder Conditions for EqualityConstrained Optimization
Because the Lagrange theorem provides firstorder necessary conditions for
constrained local optima only, the solution to the firstorder conditions can
be a constrained local maximum, constrained local minimum or neither.
We will characterize secondorder sufficient conditions for constraint local
optima.
Theorem.
Consider
max
f
(
x
)
subject to
g
(
x
) =
c
Suppose that
(a)
x
*
satisfies
g
(
x
*
) =
c
,
(b) There exists
λ
*
such that (
x
*
, λ
*
) is a critical point of the Lagrangian,
(c) The Hessian of the Lagrangian with respect to
x
at (
x
*
, λ
*
),
∇
2
x
L
(
x
*
, λ
*
),
is negative definite on the linear constraint set
{
v
:
Dg
(
x
*
)
v
= 0
}
; that
is,
v
6
= 0 and
Dg
(
x
*
)
v
= 0
⇒
v
>
∇
2
x
L
(
x
*
, λ
*
)
v <
0
.
Then
x
*
is a strict local constrained max.
Theorem
. Consider
min
f
(
x
)
subject to
g
(
x
) =
c
Suppose that
(a)
x
*
satisfies
g
(
x
*
) =
c
,
(b) There exists
λ
*
such that (
x
*
, λ
*
) is a critical point of the Lagrangian,
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 Critical Point, Optimization, hessian matrix, leading principal minors

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