Engineering Optimization: Theory and Practice, Fourth Edition

288 Nonlinear Programming I: One-Dimensional Minimization Methods

Example 5.12 Find the minimum of the function

f (λ)= 0. 65 −

0. 75

1 +λ^2

− 0. 65 λtan−^1

1

λ

using the Newton–Raphson method with the starting pointλ 1 = 0. 1. Useε= 0 .01 in

Eq. (5.66) for checking the convergence.

SOLUTION The first and second derivatives of the functionf (λ)are given by

f′(λ)=

1. 5 λ

( 1 +λ^2 )^2

+

0. 65 λ

1 +λ^2

− 0. 6 5 tan−^1

1

λ

f′′(λ)=

1. 5 ( 1 − 3 λ^2 )

( 1 +λ^2 )^3

+

0. 65 ( 1 −λ^2 )

( 1 +λ^2 )^2

+

0. 65

1 +λ^2

=

2. 8 − 3. 2 λ^2

( 1 +λ^2 )^3

Iteration 1

λ 1 = 0. 1 , f (λ 1 ) =− 0. 188197 , f′(λ 1 ) =− 0. 744832 , f′′(λ 1 ) = 2. 68659

λ 2 =λ 1 −

f′(λ 1 )

f′′(λ 1 )

= 0. 377241

Convergence check:|f′(λ 2 ) = |−| 0. 138230 |>ε.

Iteration 2

f (λ 2 )=− 0. 303279 , f′(λ 2 ) =− 0. 138230 , f′′(λ 2 ) = 1. 57296

λ 3 =λ 2 −

f′(λ 2 )

f′′(λ 2 )

= 0. 465119

Convergence check:|f′(λ 3 ) = |−| 0. 0179078 |>ε.

Iteration 3

f (λ 3 )=− 0. 309881 , f′(λ 3 ) =− 0. 0179078 , f′′(λ 3 ) = 1. 17126

λ 4 =λ 3 −

f′(λ 3 )

f′′(λ 3 )

= 0. 480409

Convergence check:|f′(λ 4 ) = |−| 0. 0005033 |< ε.

Since the process has converged, the optimum solution is taken asλ∗≈λ 4 =

0. 4 80409.

5.12.2 Quasi-Newton Method

If the function being minimizedf (λ)is not available in closed form or is difficult to

differentiate, the derivativesf′(λ) andf′′(λ) n Eq. (5.65) can be approximated by thei