Engineering Optimization: Theory and Practice, Fourth Edition

364 Nonlinear Programming II: Unconstrained Optimization Techniques

2.A quadratic function:

f (x 1 , x 2 ) =(x 1 + 2 x 2 − 7 )^2 +( 2 x 1 +x 2 − 5 )^2 (6.141)

X 1 =

{

0

0

}

, X∗=

{

1

3

}

f 1 = 7. 40 , f∗= 0. 0

3 .Powell’s quartic function [6.7]:

f (x 1 , x 2 , x 3 , x 4 )=(x 1 + 01 x 2 )^2 + 5 (x 3 −x 4 )^2

+ (x 2 − 2 x 3 )^4 + 01 (x 1 −x 4 )^4 (6.142)

X 1 T={x 1 x 2 x 3 x 4 } 1 = { 3 − 1 0 1}, X∗T= { 0 0 0 0}

f 1 = 152. 0 , f∗= 0. 0

4 .Fletcher and Powell’s helical valley [6.21]:

f (x 1 , x 2 , x 3 ) = 100

{

[x 3 − 01 θ (x 1 , x 2 )]^2 +[

√

x 12 +x 22 − ] 12

}

+x 32 (6.143)

where

2 π θ (x 1 , x 2 )=











arctan

x 2

x 1

ifx 1 > 0

π+arctan

x 2

x 1

ifx 1 < 0

X 1 =







− 1

0

0







, X∗=







1

0

0







f 1 = 52 , 000. 0 , f∗= 0. 0

5 .A nonlinear function of three variables [6.7]:

f (x 1 , x 2 , x 3 )=

1

1 +(x 1 −x 2 )^2

+ ins

(

1

2

π x 2 x 3

)

+exp

[

−

(

x 1 +x 3

x 2

− 2

) 2 ]

(6.144)

X 1 =







0

1

2







, X∗=







1

1

1







f 1 = 1. 5 , f∗=fmax= 3. 0

6 .Freudenstein and Roth function [6.27]:

f (x 1 , x 2 )={− 13 +x 1 +[( 5 −x 2 )x 2 − ] 2 x 2 }^2

+ {− 29 +x 1 + [(x 2 + 1 )x 2 − 4] 1 x 2 }^2 (6.145)