Geoscience Reference
In-Depth Information
2
3
!
λ
!
!
X
X
n
R
X
∂
C
p
x
ðÞ
∂
k
P
k
X
q∈P
k
1
P
k
X
q∈P
k
X
a∈L
ˉ
q
g
a
ʴ
aq
ʴ
ap
4
5
x
q
k
x
q
x
p
þλ
k¼
1
p∈P
k
q∈P
2
4
3
5
z
p
z
p
h
i
h
i
þ
X
n
R
X
∂ʳ
k
z
ðÞ
∂
x
p
x
p
z
p
ˉ
p
k¼
1
p∈P
k
"
#
h
i
0,
þ
X
X
T
kp
þz
p
X
q
X
n
R
g
a
x
q
ʴ
aq
ʴ
ap
ˉ
p
ˉ
p
8 x
ð
;
z
;ˉ
Þ∈
K
,
k¼
1
p
∈P
k
∈P
a
∈
L
ð
15
:
24
Þ
where
∂
C
p
ðÞ
∂
x
p
X
a∈L
∂
^
c
a
f
ðÞ
∂
ʴ
ap
,
8p
∈P
k
;
k ¼
1,
...
,
n
R
:
ð
15
:
25
Þ
f
a
Proof:
Consider the optimization formulation in (
15.21
). The convexity of the
objective function follows, under the imposed assumptions, from Nagurney
et al. (
2012a
), and since the
ʳ
k
(
z
) functions are assumed to be convex.
According to Bertsekas and Tsitsiklis (
1989
) (page 287), the optimization
problem (
15.21
), along with its inequality constraints (
15.19
), is equivalent to the
below inequality which is resulted from the Karush-Kuhn-Tucker
(KKT)
(cf. Karush
1939
; Kuhn and Tucker
1951
) conditions:
2
3
X
n
R
X
þ
X
q∈P
X
a∈L
ˉ
q
g
a
ʴ
aq
ʴ
ap
Δ
k
þ λ
k
E
Δ
k
þ ʳ
k
zðÞ
∂
∂
4
x
p
C
p
xðÞþλ
k
E
5
k¼
1
p∈P
k
h
i
x
p
x
p
2
3
h
i
þ
X
n
R
X
k
þ λ
k
þ ʳ
k
zðÞ
ˉ
p
∂
∂
4
z
p
C
p
xðÞþλ
5
z
p
z
p
k
E
k
E
Δ
Δ
k¼
1
p∈P
k
"
#
h
i
0,
þ
X
X
T
kp
þ z
p
X
q
X
n
R
g
a
x
q
ʴ
aq
ʴ
ap
ˉ
p
ˉ
p
8 x
ð
;
z
; ˉ
Þ∈
K
:
k¼
1
p
∈P
k
∈P
a∈L
ð
15
:
26
Þ
Substituting the partial derivatives in (
15.26
), and using (
15.25
), (
15.22a
), and
(
15.22b
), one obtains the variational inequality (
15.24
).
□
Variational inequality (
15.24
) can be put into standard form (Nagurney (
1999
))
as follows: determine
X
∗
∈K
such that:
