Engineering Optimization: Theory and Practice, Fourth Edition

7.15 Exterior Penalty Function Method 443 SUMT method is a global one. In such cases one has to satisfy with a local minimum onl ...

444 Nonlinear Programming III: Constrained Optimization Techniques Figure 7.11 Aφfunction discontinuous forq=0. Figure 7.12 Deri ...

7.15 Exterior Penalty Function Method 445 Figure 7.13 Aφfunction forq>1. 4.q>1. Theφfunction will have continuous first de ...

446 Nonlinear Programming III: Constrained Optimization Techniques Example 7.9 Minimizef (x 1 , x 2 )=^13 (x 1 + 1 )^3 +x 2 subj ...

7.16 Extrapolation Techniques in the Interior Penalty Function Method 447 Table 7.5 Results for Example 7.9 Value ofr x∗ 1 x∗ 2 ...

448 Nonlinear Programming III: Constrained Optimization Techniques X∗ 1 ,X∗ 2 ,... ,X∗kconverges to the minimum pointX∗, and the ...

7.16 Extrapolation Techniques in the Interior Penalty Function Method 449 From Eqs. (7.206) and (7.208), the extrapolated value ...

450 Nonlinear Programming III: Constrained Optimization Techniques 7.16.2 Extrapolation of the Functionf As in the case of the d ...

7.17 Extended Interior Penalty Function Methods 451 Example 7.10 Find the extrapolated values ofXandf in Example 7.8 using the r ...

452 Nonlinear Programming III: Constrained Optimization Techniques function is constructed as follows: φk= φ(X, rk) =f(X)+rk ∑m ...

7.18 Penalty Function Method for Problems with Mixed Equality and Inequality Constraints 453 Figure 7.14 Linear extended penalty ...

454 Nonlinear Programming III: Constrained Optimization Techniques Figure 7.15 Graphs ofφk. methods that can be used to solve a ...

7.18 Penalty Function Method for Problems with Mixed Equality and Inequality Constraints 455 H (rk) →∞, the quantity (^) jp= 1 l ...

456 Nonlinear Programming III: Constrained Optimization Techniques As in the case of Eq. (7.199), this function has to be minimi ...

7.19 Penalty Function Method for Parametric Constraints 457 Figure 7.17 Output angles generated and desired. Figure 7.18 Rectang ...

458 Nonlinear Programming III: Constrained Optimization Techniques Another method of handling the parametric constraints is to c ...

7.20 Augmented Lagrange Multiplier Method 459 Figure 7.19 Numerical integration procedure. whereris the number of discrete value ...

460 Nonlinear Programming III: Constrained Optimization Techniques subject to hj( X)= 0 , j= 1 , 2 ,... , p, p < n (7.240) Th ...

7.20 Augmented Lagrange Multiplier Method 461 whereX(k)denotes the starting vector used in the minimization ofA.The value ofrk i ...

462 Nonlinear Programming III: Constrained Optimization Techniques 7.20.2 Inequality-Constrained Problems Consider the following ...