Engineering Optimization: Theory and Practice, Fourth Edition

308 Nonlinear Programming II: Unconstrained Optimization Techniques

whereλiis the ith eigenvalue anduiis the corresponding eigenvector. In the present

case, the eigenvalues,λi, are given by

∣

∣

∣

∣

12 −λi − 6

− 6 4 −λi

∣

∣

∣

∣=λ

2

i−^61 λi+^21 =^0 (E^4 )

which yieldλ 1 = 8 +

√

52 = 15 .2111 andλ 2 = 8 −

√

52 = 0 .7889. The eigenvector

uicorresponding toλican be found by solving Eq. (E 3 ):

[

12 −λ 1 − 6

− 6 4 −λ 1

]{

u 11

u 21

}

=

{

0

0

}

or ( 12 −λ 1 )u 11 − 6 u 21 = 0

or u 21 = − 0. 5332 u 11

that is,

u 1 =

{

u 11

u 21

}

=

{

1. 0

− 0. 5332

}

and

[

12 −λ 2 − 6

− 6 4 −λ 2

]{

u 12

u 22

}

=

{

0

0

}

or ( 12 −λ 2 )u 12 − 6 u 22 = 0

or u 22 = 1. 8685 u 12

that is,

u 2 =

{

u 12

u 22

}

=

{

1. 0

1. 8685

}

Thus the transformation that reduces [A] to a diagonal form is given by

X=[R]Y=[u 1 u 2 ]Y=

[

1 1

− 0 .5352 1. 8685

]{

y 1

y 2

}

(E 5 )

thatis,

x 1 =y 1 +y 2

x 2 = − 0. 5352 y 1 + 1. 8685 y 2

This yields the new quadratic term as^12 YT[ A ̃]Y,where

[A ̃]=[R]T[ [A]R]=

[

19 .5682 0. 0

0. 0 3. 5432

]

and hence the quadratic function becomes

f (y 1 , y 2 )=BT[R]Y+^12 YT[A ̃]Y

= 0. 0704 y 1 − 4. 7370 y 2 +^12 ( 91. 8682 )y 12 + 21 ( 4323. 5 )y^22 (E 6 )