Engineering Optimization: Theory and Practice, Fourth Edition

5.7 Fibonacci Method 263 5.7 Fibonacci Method As stated earlier, theFibonacci methodcan be used to find the minimum of a functio ...

264 Nonlinear Programming I: One-Dimensional Minimization Methods and with one experiment left in it. This experiment will be at ...

5.7 Fibonacci Method 265 Table 5.2 Reduction Ratios Value ofn Fibonacci number,Fn Reduction ratio,Ln/L 0 0 1 1.0 1 1 1.0 2 2 0.5 ...

266 Nonlinear Programming I: One-Dimensional Minimization Methods Figure 5.9 Flowchart for implementing Fibonacci search method. ...

5.8 Golden Section Method 267 SOLUTION Heren=6 andL 0 = 3. 0 , which yield L∗ 2 = Fn− 2 Fn L 0 = 5 13 ( 3. 0 )= 1. 153846 Thus t ...

268 Nonlinear Programming I: One-Dimensional Minimization Methods Figure 5.10 Graphical representation of the solution of Exampl ...

5.8 Golden Section Method 269 Figure 5.10 (continued) This result can be generalized to obtain Lk= iml N →∞ ( FN− 1 FN )k− 1 L 0 ...

270 Nonlinear Programming I: One-Dimensional Minimization Methods Eq. (5.21) can be expressed as γ≃ 1 γ + 1 that is, γ^2 −γ− 1 = ...

5.9 Comparison of Elimination Methods 271 Example 5.8 Minimize the function f (x)= 0. 65 −[0. 75 /( 1 +x^2 ) ]− 0. 65 xtan−^1 ( ...

272 Nonlinear Programming I: One-Dimensional Minimization Methods Table 5.3 Final Intervals of Uncertainty Method Formula n= 5 n ...

5.10 Quadratic Interpolation Method 273 SOLUTION The new design pointXcan be expressed as X= { x 1 x 2 } =X 1 +λS= { − 2 +λ − 2 ...

274 Nonlinear Programming I: One-Dimensional Minimization Methods that is, λ ̃∗= −b 2 c (5.30) The sufficiency condition for the ...

5.10 Quadratic Interpolation Method 275 provided that c= fC+fA− 2 fB 2 t^2 > 0 (5.41) The inequality (5.41) can be satisfied ...

276 Nonlinear Programming I: One-Dimensional Minimization Methods l l l l~* l~* l~* Figure 5.13 Possible outcomes when the funct ...

5.10 Quadratic Interpolation Method 277 df/dλand use the criterion ∣ ∣ ∣ ∣ ∣ f (λ ̃∗+ λ ̃∗) −f(λ ̃∗− λ ̃∗) 2 λ ̃∗ ∣ ∣ ∣ ∣ ∣ ≤ε ...

278 Nonlinear Programming I: One-Dimensional Minimization Methods Table 5.5 Refitting Scheme New points for refitting Case Chara ...

5.10 Quadratic Interpolation Method 279 and h(λ ̃∗ )=h( 1. 135 )= 5 − 204 ( 1. 135 )+ 90 ( 1. 135 )^2 = − 110. 9 Since f ̃=f (λ ...

280 Nonlinear Programming I: One-Dimensional Minimization Methods 5.11 Cubic Interpolation Method The cubic interpolation method ...

5.11 Cubic Interpolation Method 281 is used to approximate the functionf (λ)between pointsAandB, we need to find the valuesfA= f ...

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′+ ...