428 Nonlinear Programming III: Constrained Optimization Techniques
By using quadratic interpolation technique (unrestricted search method can also be used
for simplicity), we find thatφattains its minimum value of 1.48 atα∗= 46 .93, which
corresponds to the new design vectorX 2 =
{
8. 7657
8. 8719
}
withf (X 2 ) = 1. 3 8874 andg 1 (X 2 ) =+ 0 .0074932 (violated slightly). Next we update
the matrix [H] using Eq. (7.143) withL ̃= 0. 1 x 1 + 0. 05773 x 2 + 21. 2450(
0. 6
x 1+
0. 3464
x 2− 0. 1
)
∇xL ̃=
∂L ̃
∂x 1
∂L ̃
∂x 2
with∂L ̃
∂x 1= 0. 1 −
7. 3470
x 12and∂L ̃
∂x 2= 0. 05773 −
4. 2417
x^22P 1 =X 2 −X 1 =
{
− 3. 1108
1. 8719
}
Q 1 = ∇xL ̃(X 2 ) −∇xL ̃(X 1 )={
0. 00438
0. 00384
}
−
{
0. 04791
− 0. 02883
}
=
{
− 0. 04353
0. 03267
}
PT 1 [H 1 ]P 1 = 31. 1811 , PT 1 Q 1 = 0. 19656
This indicates thatPT 1 Q 1 < 0. 2 PT 1 [H 1 ]P 1 , and henceθiscomputed using Eq. (7.147) asθ=( 0. 8 )( 13. 1811 )
13. 1811 − 0. 19656
= 0. 81211
γ=θQ 1 + ( 1 −θ)[H 1 ]P 1 ={
0. 54914
− 0. 32518
}
Hence
[H 2 ]=[
0. 2 887 0. 4283
0 .4283 0. 7422
]
We can now start another iteration by defining a new quadratic programming problem
using Eq. (7.138) and continue the procedure until the optimum solution is found.
Note that the objective function reduced from a value of 1.5917 to 1.38874 in one
iteration whenXchanged fromX 1 toX 2.INDIRECT METHODS
7.11 Transformation Techniques
If the constraintsgj( X)areexplicit functions of the variablesxiand have certain simple
forms, it may be possible to make a transformation of the independent variables such