Engineering Optimization: Theory and Practice, Fourth Edition

6.1 Introduction 303 Figure 6.2 Finite-element model of a cantilever beam. of the beam(F ), which can be expressed as [6.1] F= 1 ...

304 Nonlinear Programming II: Unconstrained Optimization Techniques Equation (E 6 ) can be rewritten as F= 1 2 ∫ 1 0 EI ( d^2 w ...

6.1 Introduction 305 those requiring only first derivatives of the function are calledfirst-order methods; those requiring both ...

306 Nonlinear Programming II: Unconstrained Optimization Techniques design variables changes the condition number†of the Hessian ...

6.1 Introduction 307 the second-order Taylor’s series approximation of a general nonlinear function at the design vectorXican be ...

308 Nonlinear Programming II: Unconstrained Optimization Techniques whereλiis the ith eigenvalue anduiis the corresponding eigen ...

6.2 Random Search Methods 309 Stage 2: Reducing [A ̃] to a Unit Matrix The transformation is given byY=[S]Z, where [S]=     ...

310 Nonlinear Programming II: Unconstrained Optimization Techniques Figure 6.3 Contours of the original and transformed function ...

6.2 Random Search Methods 311 Figure 6.3 (continued). 6.2.1 Random Jumping Method Although the problem is an unconstrained one, ...

312 Nonlinear Programming II: Unconstrained Optimization Techniques 6.2.2 Random Walk Method Therandom walk methodis based on ge ...

6.2 Random Search Methods 313 Table 6.2 Minimization offby Random Walk Method Step Number of Current objective length, trials Co ...

314 Nonlinear Programming II: Unconstrained Optimization Techniques 6.2.4 Advantages of Random Search Methods 1.These methods ca ...

6.4 Univariate Method 315 variables (n=10), the number of grid points will be 3^10 = 95 ,049 withpi= and 3 410 = 1 , 048 ,576 wi ...

316 Nonlinear Programming II: Unconstrained Optimization Techniques The univariate method is very simple and can be implemented ...

6.4 Univariate Method 317 Step 3: To find whether the value off decreases alongS 1 or−S 1 , we use the probe lengthε. Since f 1 ...

318 Nonlinear Programming II: Unconstrained Optimization Techniques Next we set the iteration number asi=3, and continue the pro ...

6.6 Powell’s Method 319 denotes the number of design variables and then searches for the minimum along the pattern directionSi, ...

320 Nonlinear Programming II: Unconstrained Optimization Techniques and hence ∇Q(X 1 ) −∇Q(X 2 )=A(X 1 −X 2 ) (6.27) IfSis any v ...

6.6 Powell’s Method 321 different starting pointsXaandXb, respectively, the line(X 1 −X 2 ) ill be conjugatew to the search dire ...

322 Nonlinear Programming II: Unconstrained Optimization Techniques whereλ∗i is found by minimizingQ(Xi+λiSi) thatso † STi∇Q(Xi+ ...