Engineering Optimization: Theory and Practice, Fourth Edition

356 Nonlinear Programming II: Unconstrained Optimization Techniques

The quantitySTi+ 1 [A]Sican be written as

STi+ 1 [A]Si= −([Bi+ 1 ]∇fi+ 1 )T[A]Si

= −∇fiT+ 1 [Bi+ 1 ][A]Si= −∇fiT+ 1 Si= 0 (E 11 )

sinceλ∗i is the minimizing step in the directionSi. Equation (E 11 ) proves that the

successive directions generated in the DFP method are [A]-conjugate and hence the

method is a conjugate gradient method.

Example 6.14 Minimizef (x 1 , x 2 )= 100 (x^21 −x 2 )^2 +( 1 −x 1 )^2 takingX 1 =

{− 2

− 2

}

as

the starting point. Use cubic interpolation method for one-dimensional minimization.

SOLUTION Since this method requires the gradient off, we find that

∇f=

{

∂f /∂x 1

∂f /∂x 2

}

=

{

400 x 1 (x 12 −x 2 )− 2 ( 1 −x 1 )

− 200 (x 12 −x 2 )

}

Iteration 1

We take

[B 1 ]=

[

10

0 1

]

AtX 1 =

{− 2

− 2

}

,∇f 1 = ∇ f(X 1 )=

{− 8064

− 1200

}

andf 1 = 609. Therefore, 3

S 1 = −[B 1 ]∇f 1 =

{

4806

1200

}

By normalizing, we obtain

S 1 =

1

[( 4806 )^2 + 200 ( 1 )^2 ]^1 /^2

{

4806

1200

}

=

{

0. 970

0. 244

}

To findλ∗i, we minimize

f(X 1 +λ 1 S 1 ) =f(− 2 + 0. 970 λ 1 , − 2 + 0. 244 λ 1 )

= 100 ( 6 − 4. 124 λ 1 + 0. 938 λ^21 )^2 + ( 3 − 0. 97 λ 1 )^2 (E 1 )

with respect toλ 1. Equation (E 1 ) gives

df

dλ 1

= 002 ( 6 − 4. 124 λ 1 + 0. 938 λ^21 )( 1. 876 λ 1 − 4. 124 )− 1. 94 ( 3 − 0. 97 λ 1 )

Since the solution of the equationdf/dλ 1 = cannot be obtained in a simple manner, 0

we use the cubic interpolation method for findingλ∗i.