Engineering Optimization: Theory and Practice, Fourth Edition

6.6 Powell’s Method 323 Example 6.5 Consider the minimization of the function f (x 1 , x 2 )= 6 x 12 + 2 x 22 − 6 x 1 x 2 −x 1 − ...

324 Nonlinear Programming II: Unconstrained Optimization Techniques Figure 6.8 Progress of Powell’s method. is stored as the vec ...

6.6 Powell’s Method 325 l l l l l l l l l Figure 6.9 Flowchart for Powell’s Method. direction are points that are minima alongSn ...

326 Nonlinear Programming II: Unconstrained Optimization Techniques Sn,S(p^1 ),S(p^2 ),... areA-conjugate. Since, by Theorem 6.2 ...

6.6 Powell’s Method 327 Asdf/dλ=0 atλ∗=^12 , we haveX 2 =X 1 +λ∗S 2 = { 0 0. 5 } . Next we minimizefalongS 1 = { 1 0 } fromX 2 = ...

328 Nonlinear Programming II: Unconstrained Optimization Techniques If we do not recognizeX 5 as the optimum point at this stage ...

6.7 Simplex Method 329 Figure 6.10 Reflection. by rejecting the vertex corresponding to the highest function value. Since the di ...

330 Nonlinear Programming II: Unconstrained Optimization Techniques X 0 is the centroid of all the pointsXiexcepti=h: X 0 = 1 n ...

6.7 Simplex Method 331 Whenever such situation is encountered, we reject the vertex corresponding to the second worst value inst ...

332 Nonlinear Programming II: Unconstrained Optimization Techniques toXeusing the relation Xe=γXr+ ( 1 −γ)X 0 (6.53) whereγis ca ...

6.7 Simplex Method 333 SOLUTION Iteration 1 Step 1: The function value at each of the vertices of the current simplex is given b ...

334 Nonlinear Programming II: Unconstrained Optimization Techniques Iteration 2 Step 1: Asf (X 1 ) = 80. 0 , f (X 2 ) = 56 .75, ...

6.8 Gradient of a Function 335 INDIRECT SEARCH (DESCENT) METHODS 6.8 Gradient of a Function The gradient of a function is ann-co ...

336 Nonlinear Programming II: Unconstrained Optimization Techniques Since the gradient vector represents the direction of steepe ...

6.8 Gradient of a Function 337 Eq. (6.61) can be rewritten as df ds = ||∇f|| ||u||cosθ (6.62) where||∇f||and||u||denote the leng ...

338 Nonlinear Programming II: Unconstrained Optimization Techniques This formula requires two additional function evaluations fo ...

6.9 Steepest Descent (Cauchy) Method 339 wherexjis the jth component ofX. But ∂xj ∂λ = ∂ ∂λ (xij+ λsij)=sij (6.66) wherexijandsi ...

340 Nonlinear Programming II: Unconstrained Optimization Techniques SOLUTION Iteration 1 The gradient offis given by ∇f= { ∂f/∂x ...

6.10 Conjugate Gradient (Fletcher–Reeves) Method 341 As f (X 3 +λ 3 S 3 )=f(− 0. 8 − 0. 2 λ 3 , 1. 2 + 0. 2 λ 3 ) = 0. 04 λ^23 − ...

342 Nonlinear Programming II: Unconstrained Optimization Techniques We have seen that Powell’s conjugate direction method requir ...