8.4 Solution Using Differential Calculus 495Equations (8.7) are called theorthogonality conditions and Eq. (8.9) is called the
normality condition. To obtain the minimum value of the objective functionf∗, the
following procedure can be adopted. Consider
f∗= (f∗)^1 = (f∗)N
j= 1
∗j=(f∗)∗ (^1) (f∗)∗ (^2) ·· ·(f∗)∗N (8.10)
Since
f∗=
U 1 ∗
∗ 1
=
U 2 ∗
∗ 2
= · · · =
UN∗
∗N
(8.11)
from Eq. (8.8), Eq. (8.10) can be rewritten as
f∗=(
U 1 ∗
∗ 1
)∗ 1 (
U 2 ∗
∗ 2
)∗ 2
·· ·
(
UN∗
∗N
)∗N
(8.12)
By substituting the relation
Uj∗=cj∏ni= 1(xi∗)aij, j= 1 , 2 ,... , NEq. (8.12) becomes
f∗=
(
c 1
∗ 1)∗
1[n
∏i= 1(x∗i)ai^1]∗
1
(
c 2
∗ 2)∗
2[n
∏i= 1(xi∗)ai^2]∗
2
·· ·
(
cN
∗N)∗N[∏ni= 1(xi∗)aiN]∗N
=
∏N
j= 1(
cj
∗j)∗j
∏N
j= 1[n
∏i= 1(x∗i)aij]∗j
=
∏N
j= 1(
cj
∗j)∗j
[n
∏i= 1(xi∗)∑N
j= 1 aij∗j]
=
∏N
j= 1(
cj
∗j)∗j
(8.13)since
∑N
j= 1aij∗j= 0 for anyi from Eq. (8.7)Thus the optimal objective functionf∗can be found from Eq. (8.13) once∗j are
determined. To determine∗j ( j= 1 , 2 ,... , N), Eqs. (8.7) and (8.9) can be used. It
can be seen that there aren+1 equations inNunknowns. IfN=n+1, there will
be as many linear simultaneous equations as there are unknowns and we can find a
unique solution.