Engineering Optimization: Theory and Practice, Fourth Edition

6.14 Davidon–Fletcher–Powell Method 355

this involves more computational effort. Another possibility is to specify a maximum
number of refits in the one-dimensional minimization method and to skip the updating
of [Bi] ifλ∗i could not be found accurately in the specified number of refits. The last
possibility is to continue updating the matrix [Bi] using the approximate values ofλ∗i
found, but restart the whole procedure after certain number of iterations, that is, restart
withi=1 in step 2 of the method.

Example 6.13 Show that the DFP method is a conjugate gradient method.

SOLUTION Consider the quadratic function

f (X)=^12 XT[A]X+BTX+C (E 1 )

for which the gradient is given by

∇f=[A]X+B (E 2 )

Equations(6.133) and (E 2 ) give

gi= ∇fi+ 1 − ∇fi=[A](Xi+ 1 −Xi) (E 3 )

Since

Xi+ 1 =Xi+λ∗iSi (E 4 )

Eq. (E 3 ) becomes

gi=λ∗i[A]Si (E 5 )

or

[A]Si=

1

λ∗i

gi (E 6 )

Premultiplication of Eq. (E 6 ) by [Bi+ 1 ] leads to

[Bi+ 1 ][A]Si=

1

λ∗i

([Bi]+[Mi]+[Ni])gi (E 7 )

Equations(6.131) and (E 5 ) yield

[Mi]gi=λ∗i

SiSTigi

STigi

=λ∗iSi (E 8 )

Equation(6.132) can be used to obtain

[Ni]gi= −

([Bi]gi)(gTi[Bi]Tgi)

gTi[Bi]gi

= −[Bi]gi (E 9 )

since [Bi] is symmetric. By substituting Eqs. (E 8 ) and (E 9 ) into Eq. (E 7 ), we obtain

[Bi+ 1 ][A]Si=

1

λ∗i

([Bi]gi+λ∗iSi−[Bi]gi)=Si (E 10 )