Engineering Optimization: Theory and Practice, Fourth Edition

6.10 Conjugate Gradient (Fletcher–Reeves) Method 343 6.10.2 Fletcher–Reeves Method The iterative procedure of Fletcher–Reeves me ...

344 Nonlinear Programming II: Unconstrained Optimization Techniques SOLUTION Iteration 1 ∇f = { ∂f/∂x 1 ∂f/∂x 2 } = { 1 + 4 x 1 ...

6.11 Newton’s Method 345 Thus the optimum point is reached in two iterations. Even if we do not know this point to be optimum, w ...

346 Nonlinear Programming II: Unconstrained Optimization Techniques Example 6.10 Show that the Newton’s method finds the minimum ...

6.11 Newton’s Method 347 Therefore, [J 1 ]−^1 = 1 4 [ + 2 − 2 −2 4 ] = [ 1 2 − 1 2 −^121 ] As g 1 = { ∂f/∂x 1 ∂f/∂x 2 } X 1 = { ...

348 Nonlinear Programming II: Unconstrained Optimization Techniques 6.12 Marquardt Method The steepest descent method reduces th ...

6.12 Marquardt Method 349 whereλ∗i is found using any of the one-dimensional search methods described in Chapter 5. Example 6.12 ...

350 Nonlinear Programming II: Unconstrained Optimization Techniques 6.13 Quasi-Newton Methods The basic iterative process used i ...

6.13 Quasi-Newton Methods 351 have been suggested in the literature for the computation of [Bi] as the iterative process progres ...

352 Nonlinear Programming II: Unconstrained Optimization Techniques using Eq. (6.111) and the new pointX 3 is determined from Eq ...

6.13 Quasi-Newton Methods 353 whereSiis the search direction,di=Xi+ 1 −Xican be rewritten as di=λ∗iSi (6.121) Thus Eq. (6.119) c ...

354 Nonlinear Programming II: Unconstrained Optimization Techniques 4.It has been shown that the BFGS method exhibits superlinea ...

6.14 Davidon–Fletcher–Powell Method 355 this involves more computational effort. Another possibility is to specify a maximum num ...

356 Nonlinear Programming II: Unconstrained Optimization Techniques The quantitySTi+ 1 [A]Sican be written as STi+ 1 [A]Si= −([B ...

6.14 Davidon–Fletcher–Powell Method 357 Cubic Interpolation Method (First Fitting) Stage 1: As the search directionS 1 is normal ...

358 Nonlinear Programming II: Unconstrained Optimization Techniques Therefore, λ ̃∗i = 2. 0 + − 113. 95 − 24. 41 + 143. 2 − 113. ...

6.14 Davidon–Fletcher–Powell Method 359 SOLUTION Iteration 1 (i= 1 ) Here ∇f 1 = ∇ f(X 1 )= { 1 + 4 x 1 + 2 x 2 − 1 + 2 x 1 + 2 ...

360 Nonlinear Programming II: Unconstrained Optimization Techniques [N 1 ] =− ([B 1 ]g 1 )([B 1 ]g 1 )T gT 1 [B 1 ]g 1 = − { − 2 ...

6.15 Broyden–Fletcher–Goldfarb–Shanno Method 361 3.Find the optimal step lengthλ∗iin the directionSiand set Xi+ 1 =Xi+λ∗iSi (6.1 ...

362 Nonlinear Programming II: Unconstrained Optimization Techniques To find the minimizing step lengthλ∗ 1 alongS 1 , we minimiz ...