Engineering Optimization: Theory and Practice, Fourth Edition

276 Nonlinear Programming I: One-Dimensional Minimization Methods

l l

l

l~* l~*

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Figure 5.13 Possible outcomes when the function is evaluated atλ=t 0 : (a) f 1 < fAand

t 0 <λ ̃∗; (b) f 1 < fAandt 0 >λ ̃∗; (c) f 1 >fAandt 0 >λ ̃∗.

f(l) f(l) f(l) f(l)

l l l l

Figure 5.14 Possible outcomes when function is evaluated atλ=t 0 and 2t 0 : (a) f 2 < f 1 and

f 2 < fA; (b) f 2 < fAandf 2 >f 1 ; (c) f 2 >fAandf 2 >f 1.

if they differ not more than by a small amount. This criterion can be stated as

∣

∣

∣

∣

∣

h(λ ̃∗) −f(λ ̃∗)

f (λ ̃∗)

∣

∣

∣

∣

∣

≤ε 1 (5.43)

Another possible test is to examine whetherdf/dλis close to zero atλ ̃∗. Since the

derivatives offare not used in this method, we can use a finite-difference formula for