6.17 MATLAB Solution of Unconstrained Optimization Problems 365
X 1 =
{
0. 5
− 2
}
, X∗=
{
5
4
}
, X∗alternate=
{
11. 41...
− 0. 8968...
}
f 1 = 004. 5 , f∗= 0. 0 , falternate∗ = 84. 9842...
7.Powell’s badly scaled function [6.28]:
f (x 1 , x 2 ) =( 10 , 000 x 1 x 2 − 1 )^2 + exp[ (−x 1 ) +exp(−x 2 ) − 1. 0 001]^2 (6.146)
X 1 =
{
0
1
}
, X∗=
{
1. 098.. .× 10 −^5
9 06. 1...
}
f 1 = 1. 1354 , f*= 0. 0
8.Brown’s badly scaled function [6.29]:
f (x 1 , x 2 ) =(x 1 − 016 )^2 + (x 2 − 2 × 10 −^6 )^2 + (x 1 x 2 − 2 )^2 (6.147)
X 1 =
{
1
1
}
, X∗=
{
106
2 × 10 −^6
}
f 1 ≈ 0112 , f∗= 0. 0
9 .Beale’s function [6.29]:
f (x 1 , x 2 ) =[ 1. 5 −x 1 ( 1 −x 2 )]^2 + 2 [. 25 −x 1 ( 1 −x^22 )]^2
+ 2 [. 625 −x 1 ( 1 −x 23 )]^2 (6.148)
X 1 =
{
1
1
}
, X∗=
{
3
0. 5
}
f 1 = 41. 203125 , f∗= 0. 0
1 0.Wood’s function [6.30]:
f (x 1 , x 2 , x 3 , x 4 )=[ 10 (x 2 −x^21 )]^2 +( 1 −x 1 )^2 + 09 (x 4 −x^23 )^2
+( 1 −x 3 )^2 + 01 (x 2 +x 4 − 2 )^2 + 0. 1 (x 2 −x 4 ) 6.149)(
X 1 =
− 3
− 1
− 3
− 1
, X∗=
1
1
1
1
f 1 = 91921. 0 , f∗= 0. 0
6.17 MATLAB Solution of Unconstrained Optimization Problems
The solution of multivariable unconstrained minimization problems using the MATLAB
functionfminuncis illustrated in this section.
Example 6.17 Find the minimum of the Rosenbrock’s parabolic valley function, given
by Eq. (6.140), starting from initial pointX 1 = {− 1. 21. 0 }T.
