Computer Aided Engineering Design
OPTIMIZATION 343 (c) Fibonacci and Golden Section Search This method determines the maximum (or minimum) of a unimodal function ...
344 COMPUTER AIDED ENGINEERING DESIGN Fibonacci search x 1 = xl + x 2 = xu – g(N) xl xu g(N)(xu – xl) g(N)(xu – xl) f(x 1 ) f(x ...
OPTIMIZATION 345 Other bracketing methods include the parabolic interpolation method wherein given three function points, a para ...
346 COMPUTER AIDED ENGINEERING DESIGN h(x) x x h(x) α α (a) | h′(xi) | < 1 (b) | h′(xi) | > 1 Figure 12.6 Convergence issu ...
OPTIMIZATION 347 At point [xi,f(xi)], a tangent can be extended to the x axis and the point of intersection of this tangent and ...
348 COMPUTER AIDED ENGINEERING DESIGN gx′ ≈ gx gx i xx i i i i () () – ( ) –1 –1 substitution of which into the Newton Raphs ...
OPTIMIZATION 349 f(X 0 + ΔX)≤f (X 0 ) for a relative maximum at X 0 as well for which reason X 0 is called a stationary point. T ...
350 COMPUTER AIDED ENGINEERING DESIGN Fig. 12.8 Examples of unconstrained optimization with two variables Consider, for example, ...
OPTIMIZATION 351 whereλi are called the Lagrangian multipliers. The Lagrangian L(X,ΛΛΛΛΛ) is treated as an unconstrained functio ...
352 COMPUTER AIDED ENGINEERING DESIGN LL L( + , + ) – ( , ) =^1 2 00 00 T (, ) + () 2 2 00 T XX X X 0 X XX X ΔΔΛΛΛΛΛΛΛΔ∂ ΛΛΔΔΛ G ...
OPTIMIZATION 353 B y A 2 1 0 –1 –2 –2 –1 0 1 2 3 x Figure 12.9 Function (thin lines) and constraint (thick line) curves for Exam ...
354 COMPUTER AIDED ENGINEERING DESIGN Fig. 12.10 Function (thin lines) and constraint (thick line) curves for Example 12.8 C D B ...
OPTIMIZATION 355 12.3.4 Karush-Kuhn-Tucker (KKT) Necessary Conditions for Optimality We may realize from above that the slack va ...
356 COMPUTER AIDED ENGINEERING DESIGN Forλ 1 and λ 2 both > 0, ST∇f may be seen always to be positive. It may be noted that ∇ ...
OPTIMIZATION 357 ∂ ∂ L x x 2 = 2 + = 0 2 λ and λ[(xx 2 + 2) – 12 ] = 0 Case I: Ifλ = 0, then x 2 = 0 and x 1 = 1 and so g 1 (1, ...
358 COMPUTER AIDED ENGINEERING DESIGN ∂ ∂ ⎡ ⎣⎢ ⎤ ⎦⎥ L x xx x 1 = 2 ( 1112 – 1) – 2 – = 2 (1 – ) 112 – 1 – 1 2 λλ λ λ = 0 ∂ ∂ L x ...
OPTIMIZATION 359 The sufficiency condition may be written as ΔXT[H]ΔX > 0 (12.28b) orH is positive definite at the optimum, w ...
360 COMPUTER AIDED ENGINEERING DESIGN the goal is to determine the optimal solution(s) among the feasible set. The system aX = b ...
OPTIMIZATION 361 xbiii = , = 1,... , ′ m ff = 0 ′ xi = 0, i = m+ 1,... , n The algorithm is intended to move from one basic feas ...
362 COMPUTER AIDED ENGINEERING DESIGN xbaas = /i′′ ′is,,, is > 0, = 1,... , i m (12.33) Since we require the largest possible ...
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