Engineering Optimization: Theory and Practice, Fourth Edition

5.11 Cubic Interpolation Method 283 whereε 1 andε 2 are small numbers whose values depend on the accuracy desired. The criterion ...

284 Nonlinear Programming I: One-Dimensional Minimization Methods Figure 5.17 Flowchart for cubic interpolation method. ...

5.11 Cubic Interpolation Method 285 Iteration 1 To find the value ofλ ̃∗and to test the convergence criteria, we first computeZa ...

286 Nonlinear Programming I: One-Dimensional Minimization Methods Thus A= 1. 84 , fA= − 41. 70 , fA′= − 13. 00 B= 2. 05 , fB= − ...

5.12 Direct Root Methods 287 Thus theNewton method, Eq. (5.65), is equivalent to using a quadratic approximation for the functio ...

288 Nonlinear Programming I: One-Dimensional Minimization Methods Example 5.12 Find the minimum of the function f (λ)= 0. 65 − 0 ...

5.12 Direct Root Methods 289 finite difference formulas as f′(λi)= f(λi+ λ)−f (λi− λ) 2 λ (5.67) f′′(λi)= f(λi+ λ)− 2 f (λi) +f ...

290 Nonlinear Programming I: One-Dimensional Minimization Methods Iteration 2 f 2 = f(λ 2 ) =− 0. 303368 , f 2 + =f(λ 2 + λ)= − ...

5.12 Direct Root Methods 291 f′(l) l A = li li+ 2 li+ 1 l* Figure 5.19 Iterative process of the secant method. the secant method ...

292 Nonlinear Programming I: One-Dimensional Minimization Methods f′(l) l l~ 1 *l ~ 2 * l~ 3 * Figure 5.20 Situation whenfA′vari ...

5.13 Practical Considerations 293 Iteration 2 Sincef′(λ 2 ) =+ 0. 0105789 >0, we set newA= 0. 4 , f′(A) =− 0. 103652 , B=λ 2 ...

294 Nonlinear Programming I: One-Dimensional Minimization Methods 5.13.3 Comparison of Methods It has been shown in Section 5.9 ...

Review Questions 295 This produces the solution or ouput as follows: x= 0.4809 fval = -0.3100 References and Bibliography 5.1 J. ...

296 Nonlinear Programming I: One-Dimensional Minimization Methods 5.10 What is a dichotomous search method? 5.11 Define the gold ...

Problems 297 5.5 The shear stress induced along thez-axis when two cylinders are in contact with each other is given by τzy pmax ...

298 Nonlinear Programming I: One-Dimensional Minimization Methods such as roller bearings, when the contact load(F )is large, a ...

Problems 299 (c)Interval halving method (d)Fibonacci method (e)Golden section method 5.14 Find the number of experiments to be c ...

300 Nonlinear Programming I: One-Dimensional Minimization Methods 5.21 Consider the problem Minimizef (X)= 100 (x 2 −x^21 )^2 +( ...

6 Nonlinear Programming II: Unconstrained Optimization Techniques 6.1 Introduction This chapter deals with the various methods o ...

302 Nonlinear Programming II: Unconstrained Optimization Techniques are satisfied. The pointX∗is guaranteed to be a relative min ...