Engineering Optimization: Theory and Practice, Fourth Edition
2.4 Multivariable Optimization with Equality Constraints 83 = ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ y 4 y 1 y 3 5 1 3 6 1 5 ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ =y 4 ( 5 − ...
84 Classical Optimization Techniques As an example, consider the problem of minimizing f (X)=f (x 1 , x 2 , x 3 ) subject to the ...
2.4 Multivariable Optimization with Equality Constraints 85 difficult task and may be prohibitive for problems with more than th ...
86 Classical Optimization Techniques choose to expressdx 1 in terms ofdx 2 , we would have obtained the requirement that (∂g/∂x ...
2.4 Multivariable Optimization with Equality Constraints 87 Necessary Conditions for a General Problem. The equations derived ab ...
88 Classical Optimization Techniques Proof: The proof is similar to that of Theorem 2.4. Notes: 1.If Q= ∑n i= 1 ∑n j= 1 ∂^2 L ∂x ...
2.4 Multivariable Optimization with Equality Constraints 89 subject to 2 π x 12 + 2 πx 1 x 2 =A 0 = 42 π The Lagrange function i ...
90 Classical Optimization Techniques L 22 = ∂^2 L ∂x 22 ∣ ∣ ∣ ∣ (X∗,λ∗) = 0 g 11 = ∂g 1 ∂x 1 ∣ ∣ ∣ ∣ (X∗,λ∗) = 4 πx 1 ∗+ 2 πx 2 ...
2.4 Multivariable Optimization with Equality Constraints 91 or db=d ̃ g= ∑n i= 1 ∂ ̃ g ∂xi dxi (2.51) Equation (2.49) can be rew ...
92 Classical Optimization Techniques 3.λ∗=. In this case, any incremental change in 0 bhas absolutely no effect on the optimum v ...
2.5 Multivariable Optimization with Inequality Constraints 93 One procedure for finding the effect onf∗of changes in the value o ...
94 Classical Optimization Techniques (necessary conditions): ∂L ∂xi (X,Y,λ)= ∂f ∂xi (X)+ ∑m j= 1 λj ∂gj ∂xi (X)= 0 , i= 1 , 2 ,. ...
2.5 Multivariable Optimization with Inequality Constraints 95 where∇f and∇gjare the gradients of the objective function and thej ...
96 Classical Optimization Techniques Figure 2.8 Feasible directionS. Example 2.12 Consider the following optimization problem: M ...
2.5 Multivariable Optimization with Inequality Constraints 97 g 2 (X)= 1 −x 1 ≤ 0 g 3 (X)= 1 −x 2 ≤ 0 g 4 (X)=x 1 − 01 ≤ 0 g 5 ( ...
98 Classical Optimization Techniques 2.5.1 Kuhn–Tucker Conditions As shown above, the conditions to be satisfied at a constraine ...
2.5 Multivariable Optimization with Inequality Constraints 99 gj≤ 0 , j= 1 , 2 ,... , m hk= 0 , k= 1 , 2 ,... , p λj≥ 0 , j= 1 , ...
100 Classical Optimization Techniques Figure 2.9 Feasible region and contours of the objective function. It is clear that∇g 1 (X ...
2.5 Multivariable Optimization with Inequality Constraints 101 Example 2.14 A manufacturing firm producing small refrigerators h ...
102 Classical Optimization Techniques that is, x 1 − 05 ≥ 0 (E 7 ) x 1 +x 2 − 001 ≥ 0 (E 8 ) x 1 +x 2 +x 3 − 501 ≥ 0 (E 9 ) λj≤ ...
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