Engineering Optimization: Theory and Practice, Fourth Edition
2 Classical Optimization Techniques 2.1 Introduction The classical methods of optimization are useful in finding the optimum sol ...
64 Classical Optimization Techniques Figure 2.1 Relative and global minima. exists as a definite number, which we want to prove ...
2.2 Single-Variable Optimization 65 Figure 2.2 Derivative undefined atx∗. 3.The theorem does not say what happens if a minimum o ...
66 Classical Optimization Techniques Theorem 2.2 Sufficient Condition Letf′(x∗)=f′′(x∗) =· · · =f(n−^1 )(x∗) = 0 , butf(n)(x∗ ) ...
2.2 Single-Variable Optimization 67 Example 2.2 In a two-stage compressor, the working gas leaving the first stage of compressio ...
68 Classical Optimization Techniques 2.3 Multivariable Optimization with No Constraints In this section we consider the necessar ...
2.5 Multivariable Optimization with Inequality Constraints viii Contents where f 1 0 − 2 =e−^2 df 1 0 − 2 =h 1 ∂f ...
70 Classical Optimization Techniques that is, f (X∗+ h)−f(X∗)=hk ∂f ∂xk (X∗)+ 1 2! d^2 f (X∗+ θh), 0 < θ < 1 Sinced^2 f (X ...
2.3 Multivariable Optimization with No Constraints 71 will have the same sign as (∂^2 f/∂xi∂xj)|X=X∗for all sufficiently smallh. ...
72 Classical Optimization Techniques Figure 2.4 Spring–cart system. SOLUTION According to the principle of minimum potential ene ...
2.3 Multivariable Optimization with No Constraints 73 The determinants of the square submatrices ofJare J 1 = ∣ ∣k 2 +k 3 ∣ ∣=k ...
74 Classical Optimization Techniques As an example, consider the functionf (x, y)=x^2 −y^2. For this function, ∂f ∂x = 2 x and ∂ ...
2.4 Multivariable Optimization with Equality Constraints 75 These equations are satisfied at the points ( 0 , 0 ), ( 0 ,−^83 ), ...
76 Classical Optimization Techniques Heremis less than or equal ton; otherwise (ifm>n), the problem becomes overdefined and, ...
2.4 Multivariable Optimization with Equality Constraints 77 The necessary conditions for the maximum off give ∂f ∂x 1 = 8 x 2 [ ...
78 Classical Optimization Techniques we indicate its salient features through the following simple problem withn=2 and m=1: Mini ...
2.4 Multivariable Optimization with Equality Constraints 79 lie on the constraint curve,g(x 1 , x 2 ) = 0. Thus any set of varia ...
80 Classical Optimization Techniques Figure 2.7 Cross section of the log. This problem has two variables and one constraint; hen ...
2.4 Multivariable Optimization with Equality Constraints 81 Necessary Conditions for a General Problem. The procedure indicated ...
82 Classical Optimization Techniques In terms of the notation of our equations, let us take the independent variables as x 3 =y ...
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