Engineering Optimization: Theory and Practice, Fourth Edition

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.