6.2 Random Search Methods 309
Stage 2: Reducing [A ̃] to a Unit Matrix
The transformation is given byY=[S]Z, where
[S]=
1
√
19. 5682
0
0
1
√
3. 5432
=
[
0 .2262 0. 0
0. 0 0. 5313
]
Stage 3: Complete Transformation
The total transformation is given by
X=[R]Y=[R][S]Z=[T]Z (E 7 )
where
[T]=[R][S]=
[
1 1
− 0 .5352 1. 8685
][
0 .2262 0
0 0. 5313
]
=
[
0 .2262 0. 5313
− 0 .1211 0. 9927
]
(E 8 )
or
x 1 = 0. 2262 z 1 + 0. 5313 z 2
x 2 = − 0. 1211 z 1 + 0. 9927 z 2
With this transformation, the quadratic function of Eq. (E 1 ) becomes
f(z 1 , z 2 )=BT[T]Z+^12 ZT[T]T[ [A]T]Z
= 0. 0160 z 1 − 2. 5167 z 2 +^12 z^21 +^12 z^22 (E 9 )
The contours of the quadratic functions given by Eqs. (E 1 ), (E 6 ), and (E 9 ) are shown
in Fig. 6.3a, b, andc, respectively.
DIRECT SEARCH METHODS
6.2 Random Search Methods
Random search methods are based on the use of random numbers in finding the min-
imum point. Since most of the computer libraries have random number generators,
these methods can be used quite conveniently. Some of the best known random search
methods are presented in this section.
