Engineering Optimization: Theory and Practice, Fourth Edition

7.7 Zoutendijk’s Method of Feasible Directions 403 subject to t 1 + 2 t 2 +α+y 1 = 3 −^43 t 1 −^43 t 2 +α+y 2 = −^83 t 1 +y 3 = ...

404 Nonlinear Programming III: Constrained Optimization Techniques 7.8 Rosen’s Gradient Projection Method The gradient projectio ...

7.8 Rosen’s Gradient Projection Method 405 is the vector of Lagrange multipliers associated with Eqs. (7.54) andβis the Lagrange ...

406 Nonlinear Programming III: Constrained Optimization Techniques IfXiis the starting point for theith iteration (at whichgj 1 ...

7.8 Rosen’s Gradient Projection Method 407 Figure 7.9 Situation whenSi=0 and someλjare negative. and this vector will be a nonze ...

408 Nonlinear Programming III: Constrained Optimization Techniques that lies outside the feasible region. Hence the following pr ...

7.8 Rosen’s Gradient Projection Method 409 value of df dλ =STi∇ f(λ) at λ=λM If the minimum value ofλ,λ∗i, lies in between λ= 0 ...

410 Nonlinear Programming III: Constrained Optimization Techniques 6.IfSi= 0 ,find the maximum step lengthλMthat is permissible ...

7.8 Rosen’s Gradient Projection Method 411 as ∇f (X 1 )= { 2 x 1 − 2 2 x 2 − 4 } X 1 = { 0 − 2 } The normalized search direction ...

412 Nonlinear Programming III: Constrained Optimization Techniques Step 7: We obtain the new pointX 2 as X 2 =X 1 +λ 1 S 1 = { 1 ...

7.9 Generalized Reduced Gradient Method 413 By adding a nonnegative slack variable to each of the inequality constraints in Eq. ...

414 Nonlinear Programming III: Constrained Optimization Techniques Consider the first variations of the objective and constraint ...

7.9 Generalized Reduced Gradient Method 415 dY=          dy 1 dy 2 .. . dyn−l          (7.100) ...

416 Nonlinear Programming III: Constrained Optimization Techniques fixed, in order to have gi( X)+dgi( X)= 0 , i= 1 , 2 ,... , m ...

7.9 Generalized Reduced Gradient Method 417 can be used for this purpose. For example, if a steepest descent method is used, the ...

418 Nonlinear Programming III: Constrained Optimization Techniques If the vectorXnewcorresponding toλ∗is found infeasible, thenY ...

7.9 Generalized Reduced Gradient Method 419 we find, atX 1 , ∇Yf=      ∂f ∂x 1 ∂f ∂x 2      X 1 = { 2 (− 2. 6 − 2 ...

420 Nonlinear Programming III: Constrained Optimization Techniques (b) The upper bound onλis given by the smaller ofλ 1 andλ 2 , ...

7.9 Generalized Reduced Gradient Method 421 Since [D]= [ ∂g 1 ∂z 1 ] =[ 4 x^33 ] =[ 4 ( 1. 02595 )^3 ] =[ 4 .319551] g 1 ( X)={− ...

422 Nonlinear Programming III: Constrained Optimization Techniques Step 2: We compute the GRG at the currentXusing Eq. (7.105). ...