2.4 Multivariable Optimization with Equality Constraints 83
=
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
y 4 y 1 y 3
5 1 3
6 1 5
∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣
=y 4 ( 5 − 3 )−y 1 ( 52 − 18 )+y 3 ( 5 − 6 )
= 2 y 4 − 7 y 1 −y 3 = 0 (E 5 )
Equations(E 4 ) nda (E 5 ) ive the necessary conditions for the minimum or the maxi-g
mum off as
y 1 =^12 y 2
y 3 = 2 y 4 − 7 y 1 = 2 y 4 −^72 y 2
(E 6 )
When Eqs.(E 6 ) re substituted, Eqs.a (E 2 ) nda (E 3 ) ake the formt
− 8 y 2 + 11 y 4 = 01
− 15 y 2 + 61 y 4 = 51
from which the desired optimum solution can be obtained as
y 1 ∗= − 745
y 2 ∗= − 375
y 3 ∗=^15574
y 4 ∗=^3037
Sufficiency Conditions for a General Problem. By eliminating the firstmvariables,
using themequality constraints (this is possible, at least in theory), the objective func-
tionfcan be made to depend only on the remaining variables,xm+ 1 , xm+ 2 ,... , xn.
Then the Taylor’s series expansion off, in terms of these variables, about the extreme
pointX∗gives
f(X∗+ dX)≃f(X∗)+
∑n
i=m+ 1
(
∂f
∂xi
)
g
dxi
+
1
2!
∑n
i=m+ 1
∑n
j=m+ 1
(
∂^2 f
∂xi∂xj
)
g
dxidxj (2.28)
where(∂f/∂xi)g is used to denote the partial derivative off with respect to xi
(holding all the other variablesxm+ 1 , xm+ 2 ,... , xi− 1 , xi+ 1 , xi+ 2 ,... , xn constant)
whenx 1 , x 2 ,... , xmare allowed to change so that the constraintsgj(X∗+ dX)= 0 ,
j= 1 , 2 ,... , m, are satisfied; the second derivative, (∂^2 f/∂xi∂xj)g, is used to denote
asimilar meaning.