Engineering Optimization: Theory and Practice, Fourth Edition

7.3 Random Search Methods 383 Figure 7.4 Relative minima introduced by constraints. DIRECT METHODS 7.3 Random Search Methods The ...

384 Nonlinear Programming III: Constrained Optimization Techniques Another procedure involves constructing an unconstrained func ...

7.4 Complex Method 385 are found one at a time by the use of random numbers generated in the range 0 to 1, as xi,j=xi(l)+ri,j(x( ...

386 Nonlinear Programming III: Constrained Optimization Techniques 4.If at any stage, the reflected pointXr (found in step 3) vi ...

7.5 Sequential Linear Programming 387 7.5 Sequential Linear Programming In thesequential linear programming(SLP)method, the solu ...

388 Nonlinear Programming III: Constrained Optimization Techniques If gj(Xi+ 1 )≤ε for j= 1 , 2 ,... , m, and |hk(Xi+ 1 ) ≤| ε, ...

7.5 Sequential Linear Programming 389 Figure 7.5 Graphical representation of the problem stated by Eq. (7.21). subject to c≤x≤d ...

390 Nonlinear Programming III: Constrained Optimization Techniques Figure 7.6 Linearization of constraint aboutc. g(c)+ dg dx (c ...

7.5 Sequential Linear Programming 391 Figure 7.7 Linearization of constraint aboute. Example 7.1 Minimizef (x 1 , x 2 )=x 1 −x 2 ...

392 Nonlinear Programming III: Constrained Optimization Techniques subject to − 2 ≤x 1 ≤ 2 − 2 ≤x 2 ≤ 2 (E 1 ) Thesolution of th ...

7.6 Basic Approach in the Methods of Feasible Directions 393 Table 7.2 Results for Example 7.1 Solution of the Iteration New lin ...

394 Nonlinear Programming III: Constrained Optimization Techniques known asmethods of feasible directions. There are many ways o ...

7.7 Zoutendijk’s Method of Feasible Directions 395 subject to ST∇gj(Xi)+θjα ≤ 0 , j= 1 , 2 ,... , p (7.30b) ST∇f+α≤ 0 (7.30c) − ...

396 Nonlinear Programming III: Constrained Optimization Techniques Eqs. (7.27) and (7.28), one would naturally be tempted to cho ...

7.7 Zoutendijk’s Method of Feasible Directions 397 s 1 ∂g 2 ∂x 1 +s 2 ∂g 2 ∂x 2 + · · · +sn ∂g 2 ∂xn +θ 2 α≤ 0 .. . s 1 ∂gp ∂x 1 ...

398 Nonlinear Programming III: Constrained Optimization Techniques t 1 ∂f ∂x 1 +t 2 ∂f ∂x 2 + · · · +tn ∂f ∂xn +α+yp+ 1 = ∑n i= ...

7.7 Zoutendijk’s Method of Feasible Directions 399 Method 1. The optimal step length,λi, can be found by any of the one-dimensio ...

400 Nonlinear Programming III: Constrained Optimization Techniques the constraintgjis active ifgj(Xi+ 1 ) = 10 −^2 , 10−^3 , 10− ...

7.7 Zoutendijk’s Method of Feasible Directions 401 that is, ε= − δ 100 |f 1 | f 1 ′ (7.45) It is to be noted that the value ofεw ...

402 Nonlinear Programming III: Constrained Optimization Techniques SOLUTION Step 1: AtX 1 = { 0 0 } : f(X 1 ) = 8 and g 1 (X 1 ) ...