Engineering Optimization: Theory and Practice, Fourth Edition

286 Nonlinear Programming I: One-Dimensional Minimization Methods

Thus

A= 1. 84 , fA= − 41. 70 , fA′= − 13. 00

B= 2. 05 , fB= − 42. 90 , fB′= 5. 35

A < λ∗< B

Iteration 3

Z=

3. 0 (− 41. 70 + 42. 90 )

( 2. 05 − 1. 84 )

− 13. 00 + 5. 35 = 9. 49

Q=[( 9. 49 )^2 + 3 ( 1. 0 )( 5. 35 )]^1 /^2 = 21. 61

Therefore,

λ ̃∗= 1. 84 +−^13.^00 +^9.^49 ±^12.^61

− 13. 00 + 5. 35 + 18. 98

( 2. 05 − 1. 84 )= 2. 0086

Convergence criterion:

f′(λ ̃∗) = 5. 0 ( 2. 0086 )^4 − 51. 0 ( 2. 0086 )^2 − 02. 0 = 0. 855

Assuming that this value is close to zero, we can stop the iterative process and take

λ∗≃λ ̃∗= 2. 0086

5.12 Direct Root Methods

The necessary condition forf (λ)to have a minimum ofλ∗is thatf′(λ∗) = 0. The

direct root methods seek to find the root (or solution) of the equation,f′(λ) = 0. Three

root-finding methods—the Newton, the quasi-Newton, and the secant methods—are

discussed in this section.

5.12.1 Newton Method

Consider the quadratic approximation of the functionf (λ)atλ=λiusing the Taylor’s

series expansion:

f (λ)=f (λi)+f′(λi)(λ−λi)+^12 f′′(λi)(λ−λi)^2 (5.63)

Bysetting the derivative of Eq. (5.63) equal to zero for the minimum off (λ), we

obtain

f′(λ)=f′(λi)+f′′(λi)(λ−λi)= 0 (5.64)

Ifλidenotes an approximation to the minimum off(λ), Eq. (5.64) can be rearranged

to obtain an improved approximation as

λi+ 1 =λi−

f′(λi)

f′′(λi)

(5.65)