Engineering Optimization: Theory and Practice, Fourth Edition

7.10 Sequential Quadratic Programming 423 The extension to include inequality constraints will be considered at a later stage. T ...

424 Nonlinear Programming III: Constrained Optimization Techniques where [∇^2 L]n×ndenotes the Hessian matrix of the Lagrange fu ...

7.10 Sequential Quadratic Programming 425 subject to gj+ ∇gjT X≤ 0 , j= 1 , 2 ,... , m hk+ ∇hTk X= 0 , k= 1 , 2 ,... , p (7.136) ...

426 Nonlinear Programming III: Constrained Optimization Techniques with λj= { |λj| j, = 1 , 2 ,... , m+pin first iteration max{| ...

7.10 Sequential Quadratic Programming 427 We assume the matrix [H 1 ] to be the identity matrix and hence the objective function ...

428 Nonlinear Programming III: Constrained Optimization Techniques By using quadratic interpolation technique (unrestricted sear ...

7.11 Transformation Techniques 429 that the constraints are satisfied automatically [7.13]. Thus it may be possible to convert a ...

430 Nonlinear Programming III: Constrained Optimization Techniques subject to x 1 +x 2 +x 3 ≤ 06 (E 2 ) x 1 ≤ 63 (E 3 ) xi≥ 0 , ...

7.12 Basic Approach of the Penalty Function Method 431 unconstrained minimization problems. Let the basic optimization problem, ...

432 Nonlinear Programming III: Constrained Optimization Techniques Figure 7.10 Penalty function methods:(a)exterior method;(b)in ...

7.13 Interior Penalty Function Method 433 approached. This behavior can also be seen from Fig. 7.10b. Thus once the uncon- strai ...

434 Nonlinear Programming III: Constrained Optimization Techniques 4.Suitable convergence criteria have to be chosen to identify ...

7.13 Interior Penalty Function Method 435 5.If all the constraints are not satisfied at the pointXM, set the new starting point ...

436 Nonlinear Programming III: Constrained Optimization Techniques 1.The relative difference between the values of the objective ...

7.13 Interior Penalty Function Method 437 effective in reducing the disparities between the contributions of the various constra ...

438 Nonlinear Programming III: Constrained Optimization Techniques Table 7.3 Results for Example 7.7 Value ofr x 1 ∗(r)=(r^1 /^2 ...

7.13 Interior Penalty Function Method 439 Table 7.4 Results for Example 7.8 Number of Starting point iterations taken for minimi ...

440 Nonlinear Programming III: Constrained Optimization Techniques Proof: IfX∗is the optimum solution of the constrained problem ...

7.13 Interior Penalty Function Method 441 By using inequalities (7.178) and (7.186), inequality (7.185) becomes f (X∗) ≤φ(X∗k, r ...

442 Nonlinear Programming III: Constrained Optimization Techniques Canceling the common terms from both sides, we can write the ...