Engineering Optimization: Theory and Practice, Fourth Edition

282 Nonlinear Programming I: One-Dimensional Minimization Methods

where

Q=(Z^2 −fA′fB′)^1 /^2 (5.55)

2 (B−A)( 2 Z+fA′+fB′)(fA′+ Z±Q)

− 2 (B−A)(fA′^2 +ZfB′ + 3 ZfA′+ 2 Z^2 )

− 2 (B+A)fA′fB′> 0 (5.56)

By specializing Eqs. (5.47) to (5.56) for the case whereA=0, we obtain

a=fA

b=fA′

c =−

1

B

(Z+fA′)

d=

1

3 B^2

( 2 Z+fA′+fB′)

λ ̃∗=B

fA′+Z±Q

fA′+fB′+ 2 Z

(5.57)

Q=(Z^2 −fA′fB′)^1 /^2 > 0 (5.58)

where

Z=

3 (fA−fB)

B

+fA′+fB′ (5.59)

The two values ofλ ̃∗in Eqs. (5.54) and (5.57) correspond to the two possibilities

for the vanishing ofh′(λ) [i.e., at a maximum ofh(λ)and at a minimum]. To avoid

imaginary values ofQ, we should ensure the satisfaction of the condition

Z^2 −fA′fB′≥ 0

in Eq. (5.55). This inequality is satisfied automatically sinceAandB are selected

such thatfA′< and 0 fB′≥. Furthermore, the sufficiency condition (when 0 A=0)

requires thatQ>0, which is already satisfied. Now we computeλ ̃∗using Eq. (5.57)

and proceed to the next stage.

Stage 4. The value of ̃λ∗found in stage 3 is the true minimum ofh(λ)and may

not be close to the minimum off (λ). Hence the following convergence criteria can be

used before choosingλ∗≈λ ̃∗:

∣

∣

∣

∣

∣

h(λ ̃∗) −f(λ ̃∗)

f (λ ̃∗)

∣

∣

∣

∣

∣

≤ε 1 (5.60)

∣

∣

∣

∣

df

dλ

∣

∣

∣

∣ ̃

λ∗

=|ST∇f|λ ̃∗|≤ε 2 (5.61)